%% This LaTeX-file was created by Wed Nov 17 14:38:47 1999 %% LyX 1.0 (C) 1995-1999 by Matthias Ettrich and the LyX Team %% Do not edit this file unless you know what you are doing. \documentclass{article} \usepackage[T1]{fontenc} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. \providecommand{\LyX}{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. \usepackage[T1]{fontenc} \makeatletter \textwidth 15.4cm \textheight 22.5cm \topmargin -1.8cm \oddsidemargin -0.0cm \renewcommand{\baselinestretch}{1.8} \newtheorem{defi}{Definition} \newtheorem{prop}{Proposition} \newtheorem{tm}{Theorem} \newtheorem{lema}{Lemma} \newtheorem{kor}{Corollary} \makeatother \makeatother \begin{document} \begin{titlepage} \vspace*{3.5cm} \par\centering \textbf{\large Enumeration of symmetric (69,17,4) designs} \\ \textbf{\large admitting Z\( _{6} \) as an automorphism group}{\large \par{}}{\large \par} \vspace*{1.6cm} \par\centering Sanja Rukavina\\ Odsjek za matematiku\\ Filozofski fakultet u Rijeci\\ Omladinska 14, 51000 Rijeka, Croatia \\ (e-mail: sanjar@mapef.pefri.hr)\par{} \vspace*{2.0cm} \textit{Running head\/:}\( Z_{6} \) acting on (69,17,4) designs \bigskip All symmetric (69,17,4) designs admitting the cyclic group of order 6 as an automorphism group are classified and their full automorphism groups are determined. \vspace*{3.0cm} \textit{Keywords\/:} symmetric design, automorphism group, orbit structure.\\ \textit{Mathematical subject classification} (1991): 05B05. \end{titlepage} \section{Introduction and preliminaries} A symmetric \( (v,k,\lambda ) \) design is a finite incidence structure \( ({\mathcal{P},\mathcal{B}},I) \), where \( {\mathcal{P}} \) and \( {\mathcal{B}} \) are disjoint sets and \( I\subseteq {\mathcal{P}}\times {\mathcal{B}} \), with the following properties: \begin{description} \item [1.]\( |{\mathcal{P}}|=|{\mathcal{B}}|=v \), \item [2.]every element of \( {\mathcal{B}} \) is incident with exactly \( k \) elements of \( {\mathcal{P}} \), \item [3.]every pair of elements of \( {\mathcal{P}} \) is incident with exactly \( \lambda \) elements of \( {\mathcal{B}} \). \end{description} According to {[}4{]}, there are 4 symmetric (69,17,4) designs known. All those designs admit an automorphism group of order 13. The aim of this article is to prove the following \begin{tm}There are 59 mutually nonisomorphic symmetric \((69,17,4)\) designs admitting the cyclic group of order 6 as an automorphism group. All of them are self-dual. Exactly 3 of those designs have full automorphism groups of order 6 isomorphic to group \( Z_{6} \) and 54 designs have full automorphism groups of order 12 isomorphic to group \( Z_{12} \). Furtermore, 2 designs have full automorphism groups of order 156; one of them is isomorphic to group \( F_{39}\times Z_{4} \) and the other is isomorphic to group \( F_{39}:Z_{4} \).\end{tm} Let \( {\mathcal{D}=(\mathcal{P},\mathcal{B}},I) \) be a symmetric \( (v,k,\lambda ) \) design and \( G\leq Aut{\mathcal{D}} \). Group \( G \) has the same number of point and block orbits. Let us denote the number of \( G- \)orbits by \( t \), point orbits by \( {\mathcal{P}}_{1},\ldots ,{\mathcal{P}}_{t} \), block orbits by \( {\mathcal{B}}_{1},\ldots ,{\mathcal{B}}_{t} \), and put \( |{\mathcal{P}}_{r}|=\omega _{r} \), \( |{\mathcal{B}}_{i}|=\Omega _{i} \). Further, denote by \( \gamma _{ir} \) the number of points of \( {\mathcal{P}}_{r} \) which are incident with the representative of the block orbit \( {\mathcal{B}}_{i} \). For these numbers the following equalities hold: \begin{eqnarray} \sum _{r=1}^{t}{\gamma }_{ir} & = & k\, ,\\ \sum _{r=1}^{t}\frac{{\Omega }_{j}}{{\omega }_{r}}{\gamma }_{ir}{\gamma }_{jr} & = & \lambda {\Omega }_{j}+{\delta }_{ij}\cdot (k-\lambda )\, . \end{eqnarray} \begin{defi}The \((t \times t)\)-matrix \(({\gamma}_{ir})\) with entries satisfying properties \((1)\) and \((2)\) is called the orbit structure for parameters \((v,k, \lambda)\) and orbit distribution \((\omega_{1}, \ldots ,\omega_{t})\), \((\Omega_{1}, \ldots ,\Omega_{t})\). \end{defi} \section{Proof of the theorem} From now on we shall denote by \( G \) an abelian group isomorphic to a cyclic group of order 6. The first step in the construction of all symmetric designs admitting \( G \) as an automorphism group is to determine all possible orbit distributions and to find all possible orbit structures related to them. Action of \( G \) is semistandard, so it is sufficient to determine point orbit distributions \( \left( \omega _{1},...,\omega _{t}\right) \). \begin{lema}Symmetric \((69,17,4)\) design admitting \( G \) as an automorphism group has one of the following orbit distributions: \begin{enumerate} \item (1,2,2,2,2,3,3,3,3,6,6,6,6,6,6,6,6), \item (1,2,3,3,3,3,6,6,6,6,6,6,6,6,6), \item (3,3,3,6,6,6,6,6,6,6,6,6,6). \end{enumerate} \end{lema} \textbf{Proof} \enspace The complete orbit structures satisfying (1) and (2) can be obtained only for orbit distributions from the statement of the lemma. \( \diamond \) Using the computer program by V. \'{C}epuli\'{c} we obtained 7 orbit structures which corresponds to the first type from lemma 1, 467 orbit structures corresponding to the second type and 257 orbit structures corresponding to the third type. Next step in construction is to ``lift'' obtained orbital structures for the group \( G\cong \left\langle \rho \mid \rho ^{6}=1\right\rangle \) to orbital structures for the cyclic group \( \left\langle \rho ^{3}\right\rangle \) with the assumption that they admit \( \rho ^{2} \) as an automorphism. That can be done only for seven structures related to the first orbit distribution from lemma 1. So, by indexing structures corresponding to the second and third type cannot be obtained symmetric (69,17,4) designs admitting cyclic group of order 6 as an automorphism group. The orbit structures related to the first type are the following.\\ ~~ OS1~\( \mid \)~~ 1~~ 2~~ 2~~ 2~~ 2~~ 3~~ 3~~ 3~~ 3~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6 --------------------------------------------------------------------------------------- ~~~ 1~~~\( \mid \)~~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 6~~ 6~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 3~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 0~~ 1~~ 3~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 3~~ 3 ~~~ 2~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 0~~ 0~~ 3~~ 3~~ 0 ~~~ 2~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 0~~ 0~~ 3 ~~~ 3~~~\( \mid \)~~~ 0~~ 2~~ 2~~ 0~~ 0~~ 1~~ 0~~ 0~~ 0~~ 0~~ 0~~ 2~~ 2~~ 2~~ 2~~ 2~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 2~~ 2~~ 4~~ 2~~ 0~~ 2~~ 0~~ 4 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 2~~ 2~~ 2~~ 0~~ 4~~ 0~~ 4~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 2~~ 2~~ 0~~ 4~~ 2~~ 4~~ 2~~ 0 ~~~ 6~~~\( \mid \)~~~ 1~~ 1~~ 0~~ 1~~ 0~~ 0~~ 1~~ 1~~ 1~~ 0~~ 2~~ 2~~ 2~~ 1~~ 1~~ 2~~ 1 ~~~ 6~~~\( \mid \)~~~ 1~~ 0~~ 1~~ 0~~ 1~~ 0~~ 1~~ 1~~ 1~~ 2~~ 0~~ 1~~ 1~~ 2~~ 2~~ 1~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 1~~ 0~~ 1~~ 2~~ 1~~ 0~~ 2~~ 1~~ 0~~ 1~~ 2~~ 2~~ 1~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 0~~ 1~~ 1~~ 1~~ 0~~ 2~~ 2~~ 1~~ 1~~ 2~~ 1~~ 0~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 0~~ 1~~ 1~~ 0~~ 2~~ 1~~ 1~~ 2~~ 2~~ 1~~ 2~~ 2~~ 0~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 1~~ 0~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 2~~ 0~~ 2~~ 2~~ 1~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 1~~ 0~~ 1~~ 0~~ 2~~ 1~~ 2~~ 1~~ 1~~ 2~~ 0~~ 1~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 0~~ 1~~ 1~~ 2~~ 1~~ 0~~ 1~~ 2~~ 2~~ 2~~ 1~~ 1~~ 2~~ 0 ~ ~ ~~ OS2~\( \mid \)~~ 1~~ 2~~ 2~~ 2~~ 2~~ 3~~ 3~~ 3~~ 3~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6 --------------------------------------------------------------------------------------- ~~~ 1~~~\( \mid \)~~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 6~~ 6~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 3~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 0~~ 1~~ 3~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 3~~ 3 ~~~ 2~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 0~~ 0~~ 3~~ 3~~ 0 ~~~ 2~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 0~~ 0~~ 3 ~~~ 3~~~\( \mid \)~~~ 0~~ 2~~ 2~~ 0~~ 0~~ 1~~ 0~~ 0~~ 0~~ 0~~ 0~~ 2~~ 2~~ 2~~ 2~~ 2~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 2~~ 2~~ 4~~ 2~~ 0~~ 2~~ 0~~ 4 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 2~~ 2~~ 2~~ 0~~ 4~~ 0~~ 4~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 2~~ 2~~ 0~~ 4~~ 2~~ 4~~ 2~~ 0 ~~~ 6~~~\( \mid \)~~~ 1~~ 1~~ 0~~ 1~~ 0~~ 0~~ 1~~ 1~~ 1~~ 0~~ 2~~ 2~~ 1~~ 2~~ 2~~ 1~~ 1 ~~~ 6~~~\( \mid \)~~~ 1~~ 0~~ 1~~ 0~~ 1~~ 0~~ 1~~ 1~~ 1~~ 2~~ 0~~ 1~~ 2~~ 1~~ 1~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 1~~ 0~~ 1~~ 2~~ 1~~ 0~~ 2~~ 1~~ 0~~ 2~~ 1~~ 1~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 0~~ 1~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 2~~ 2~~ 1~~ 0~~ 2~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 0~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 1~~ 1~~ 1~~ 2~~ 2~~ 0~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 1~~ 0~~ 1~~ 1~~ 0~~ 2~~ 2~~ 1~~ 1~~ 0~~ 2~~ 2~~ 1~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 1~~ 0~~ 1~~ 0~~ 2~~ 1~~ 1~~ 2~~ 2~~ 2~~ 0~~ 1~~ 2~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 0~~ 1~~ 1~~ 2~~ 1~~ 0~~ 1~~ 2~~ 2~~ 1~~ 2~~ 2~~ 1~~ 0 ~ ~~~ ~~ OS3~\( \mid \)~~ 1~~ 2~~ 2~~ 2~~ 2~~ 3~~ 3~~ 3~~ 3~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6 --------------------------------------------------------------------------------------- ~~~ 1~~~\( \mid \)~~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 6~~ 6~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 3~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 3~~ 0~~ 3~~ 0~~ 0~~ 3~~ 3~~ 0 ~~~ 2~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 0~~ 3~~ 0~~ 3~~ 0~~ 3 ~~~ 2~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 3~~ 3~~ 0~~ 3~~ 0~~ 3~~ 0 ~~~ 3~~~\( \mid \)~~~ 0~~ 2~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 0~~ 0~~ 2~~ 4~~ 2~~ 0~~ 2~~ 2~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 2~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 2~~ 0~~ 2~~ 2~~ 2~~ 0~~ 2~~ 4 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 2~~ 2~~ 2~~ 0~~ 4~~ 4~~ 0~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 2~~ 2~~ 0~~ 4~~ 2~~ 2~~ 4~~ 0 ~~~ 6~~~\( \mid \)~~~ 1~~ 1~~ 1~~ 1~~ 0~~ 0~~ 1~~ 1~~ 1~~ 0~~ 1~~ 1~~ 2~~ 2~~ 2~~ 1~~ 1 ~~~ 6~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 0~~ 1~~ 1~~ 0~~ 1~~ 1~~ 1~~ 2~~ 1~~ 1~~ 1~~ 1~~ 2~~ 3 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 1~~ 0~~ 1~~ 2~~ 1~~ 1~~ 0~~ 1~~ 1~~ 0~~ 1~~ 2~~ 2~~ 2~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 1~~ 0~~ 1~~ 1~~ 0~~ 2~~ 2~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 0~~ 1~~ 0~~ 1~~ 2~~ 1~~ 2~~ 1~~ 2~~ 2~~ 2~~ 0~~ 1~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 1~~ 0~~ 1~~ 0~~ 2~~ 1~~ 2~~ 1~~ 2~~ 1~~ 0~~ 2~~ 2~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 0~~ 1~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 2~~ 2~~ 1~~ 2~~ 0~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 1~~ 0~~ 1~~ 2~~ 1~~ 0~~ 1~~ 3~~ 1~~ 2~~ 1~~ 1~~ 1~~ 2 ~ ~~~ ~~ OS4~\( \mid \)~~ 1~~ 2~~ 2~~ 2~~ 2~~ 3~~ 3~~ 3~~ 3~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6 --------------------------------------------------------------------------------------- ~~~ 1~~~\( \mid \)~~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 6~~ 6~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 3~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 3~~ 0~~ 3~~ 0~~ 0~~ 3~~ 3~~ 0 ~~~ 2~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 0~~ 3~~ 0~~ 3~~ 0~~ 3 ~~~ 2~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 3~~ 3~~ 0~~ 3~~ 0~~ 3~~ 0 ~~~ 3~~~\( \mid \)~~~ 0~~ 2~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 0~~ 0~~ 2~~ 2~~ 2~~ 2~~ 4~~ 2~~ 0 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 2~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 2~~ 0~~ 0~~ 2~~ 4~~ 2~~ 2~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 2~~ 2~~ 4~~ 0~~ 2~~ 2~~ 0~~ 4 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 2~~ 2~~ 2~~ 4~~ 0~~ 0~~ 4~~ 2 ~~~ 6~~~\( \mid \)~~~ 1~~ 1~~ 1~~ 1~~ 0~~ 0~~ 1~~ 1~~ 1~~ 0~~ 1~~ 2~~ 2~~ 1~~ 1~~ 1~~ 2 ~~~ 6~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 0~~ 1~~ 1~~ 0~~ 1~~ 1~~ 1~~ 2~~ 0~~ 1~~ 2~~ 2~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 1~~ 0~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 0~~ 2~~ 1~~ 1~~ 1~~ 2~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 1~~ 0~~ 1~~ 1~~ 0~~ 2~~ 2~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 0~~ 1~~ 1~~ 2~~ 1~~ 0~~ 1~~ 2~~ 1~~ 2~~ 2~~ 0~~ 1~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 1~~ 0~~ 2~~ 1~~ 1~~ 0~~ 1~~ 2~~ 1~~ 1~~ 0~~ 2~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 0~~ 1~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 2~~ 2~~ 1~~ 2~~ 0~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 1~~ 2~~ 1~~ 2~~ 2~~ 1~~ 2~~ 2~~ 2~~ 1~~ 0 ~ ~~~ ~~ OS5~\( \mid \)~~ 1~~ 2~~ 2~~ 2~~ 2~~ 3~~ 3~~ 3~~ 3~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6 --------------------------------------------------------------------------------------- ~~~ 1~~~\( \mid \)~~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 6~~ 6~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 3~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 3~~ 3~~ 0~~ 3~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 0~~ 0~~ 3~~ 3~~ 0 ~~~ 2~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 0~~ 0~~ 3 ~~~ 3~~~\( \mid \)~~~ 0~~ 2~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 2~~ 0~~ 2~~ 2~~ 2~~ 2~~ 0~~ 4 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 2~~ 0~~ 0~~ 1~~ 0~~ 0~~ 0~~ 0~~ 2~~ 4~~ 0~~ 2~~ 2~~ 2~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 2~~ 2~~ 2~~ 4~~ 0~~ 0~~ 4~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 2~~ 2~~ 0~~ 2~~ 4~~ 4~~ 2~~ 0 ~~~ 6~~~\( \mid \)~~~ 1~~ 1~~ 0~~ 1~~ 0~~ 1~~ 0~~ 1~~ 1~~ 0~~ 2~~ 1~~ 1~~ 1~~ 2~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 1~~ 0~~ 1~~ 0~~ 1~~ 0~~ 1~~ 1~~ 1~~ 2~~ 0~~ 1~~ 1~~ 2~~ 1~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 1~~ 1~~ 0~~ 0~~ 1~~ 2~~ 1~~ 1~~ 1~~ 2~~ 2~~ 2~~ 1~~ 1~~ 0 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 1~~ 0~~ 1~~ 2~~ 1~~ 1~~ 0~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 2~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 0~~ 1~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 2~~ 1~~ 2~~ 0~~ 2~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 1~~ 0~~ 1~~ 1~~ 0~~ 2~~ 2~~ 1~~ 1~~ 2~~ 0~~ 2~~ 1~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 1~~ 0~~ 1~~ 2~~ 1~~ 0~~ 2~~ 2~~ 1~~ 0~~ 2~~ 1~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 2~~ 2~~ 1~~ 1~~ 2~~ 0~~ 2 ~ ~~~ ~~ OS6~\( \mid \)~~ 1~~ 2~~ 2~~ 2~~ 2~~ 3~~ 3~~ 3~~ 3~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6 --------------------------------------------------------------------------------------- ~~~ 1~~~\( \mid \)~~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 6~~ 6~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 3~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 3~~ 3~~ 0~~ 3~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 0~~ 0~~ 3~~ 3~~ 0 ~~~ 2~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 0~~ 0~~ 3 ~~~ 3~~~\( \mid \)~~~ 0~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 2~~ 4~~ 2~~ 2~~ 0~~ 2~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 2~~ 0~~ 2~~ 4~~ 0~~ 2~~ 2~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 0~~ 2~~ 2~~ 2~~ 0~~ 2~~ 4~~ 0~~ 4 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 2~~ 2~~ 0~~ 2~~ 4~~ 2~~ 4~~ 0 ~~~ 6~~~\( \mid \)~~~ 1~~ 1~~ 0~~ 0~~ 1~~ 1~~ 0~~ 1~~ 1~~ 1~~ 1~~ 0~~ 2~~ 1~~ 2~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 1~~ 0~~ 1~~ 1~~ 0~~ 0~~ 1~~ 1~~ 1~~ 1~~ 1~~ 2~~ 0~~ 2~~ 1~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 1~~ 1~~ 0~~ 2~~ 1~~ 0~~ 1~~ 0~~ 2~~ 1~~ 1~~ 1~~ 2~~ 2~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 1~~ 0~~ 1~~ 1~~ 2~~ 1~~ 0~~ 2~~ 0~~ 1~~ 1~~ 2~~ 1~~ 1~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 1~~ 0~~ 0~~ 1~~ 1~~ 2~~ 2~~ 1~~ 1~~ 2~~ 2~~ 2~~ 0~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 0~~ 1~~ 1~~ 0~~ 2~~ 1~~ 1~~ 2~~ 2~~ 1~~ 2~~ 2~~ 1~~ 0 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 1~~ 0~~ 1~~ 1~~ 2~~ 0~~ 2~~ 2~~ 1~~ 2~~ 0~~ 1~~ 2~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 1~~ 1~~ 1~~ 0~~ 2~~ 2~~ 2~~ 2~~ 1~~ 1~~ 0~~ 2~~ 2 ~ ~~~ ~~ OS7~\( \mid \)~~ 1~~ 2~~ 2~~ 2~~ 2~~ 3~~ 3~~ 3~~ 3~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6~~ 6 --------------------------------------------------------------------------------------- ~~~ 1~~~\( \mid \)~~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 6~~ 6~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 3~~ 0~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 1~~ 0~~ 0~~ 0~~ 1~~ 0~~ 3~~ 0~~ 0~~ 0~~ 3~~ 3~~ 3~~ 0~~ 3~~ 0~~ 0 ~~~ 2~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 0~~ 0~~ 3~~ 3~~ 0 ~~~ 2~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 2~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 3~~ 0~~ 3~~ 3~~ 0~~ 0~~ 3 ~~~ 3~~~\( \mid \)~~~ 0~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 2~~ 2~~ 2~~ 4~~ 2~~ 2~~ 0 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 2~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 2~~ 0~~ 2~~ 2~~ 2~~ 4~~ 0~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 0~~ 2~~ 2~~ 0~~ 4~~ 0~~ 2~~ 4~~ 2 ~~~ 3~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 0~~ 2~~ 2~~ 4~~ 0~~ 2~~ 0~~ 2~~ 4 ~~~ 6~~~\( \mid \)~~~ 1~~ 1~~ 0~~ 0~~ 1~~ 1~~ 0~~ 1~~ 1~~ 1~~ 1~~ 1~~ 1~~ 1~~ 2~~ 1~~ 3 ~~~ 6~~~\( \mid \)~~~ 1~~ 0~~ 1~~ 1~~ 0~~ 0~~ 1~~ 1~~ 1~~ 1~~ 1~~ 1~~ 1~~ 2~~ 1~~ 3~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 1~~ 1~~ 0~~ 1~~ 1~~ 2~~ 0~~ 1~~ 1~~ 2~~ 2~~ 0~~ 1~~ 1~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 1~~ 0~~ 1~~ 1~~ 1~~ 0~~ 2~~ 1~~ 1~~ 2~~ 2~~ 1~~ 0~~ 2~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 1~~ 0~~ 1~~ 0~~ 1~~ 2~~ 0~~ 1~~ 2~~ 1~~ 0~~ 1~~ 2~~ 2~~ 1~~ 2 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 1~~ 0~~ 1~~ 2~~ 1~~ 1~~ 0~~ 1~~ 2~~ 1~~ 0~~ 2~~ 2~~ 2~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 1~~ 0~~ 1~~ 0~~ 1~~ 2~~ 1~~ 3~~ 2~~ 1~~ 1~~ 2~~ 1~~ 1 ~~~ 6~~~\( \mid \)~~~ 0~~ 0~~ 0~~ 0~~ 1~~ 0~~ 1~~ 2~~ 1~~ 3~~ 1~~ 1~~ 2~~ 2~~ 1~~ 1~~ 1 ~ The numbers of obtained new orbital structures for the cyclic group \( \left\langle \rho ^{3}\right\rangle \) are as follows: OS1\( \rightarrow \)51, OS2\( \rightarrow \)51, OS3\( \rightarrow \)10, OS4\( \rightarrow \)49, OS5\( \rightarrow \)10, OS6\( \rightarrow \)23 and OS7\( \rightarrow \)7. We shall proceed by indexing obtained structures, that is by finding all possible (69,17,4) designs that can be constructed from those structures. That can be done only for some of them and the numbers of such structures are the following: OS1\( \rightarrow \)27, OS2\( \rightarrow \)27, OS3\( \rightarrow \)0, OS4\( \rightarrow \)3, OS5\( \rightarrow \)0, OS6 \( \rightarrow \)0 and OS7\( \rightarrow \)2. From each structure that can be indexing we get 8 mutually isomorphic designs. Designs obtained from different structures are mutually nonisomorphic. So, we have 59 nonisomorphic designs. Using the computer program by V. \'{C}epuli\'{c} we have found out that all obtained designs are self-dual. They are numbered as follows: OS1\( \rightarrow \)1-27, OS2\( \rightarrow \)28-54, OS4\( \rightarrow \)55-57, OS7\( \rightarrow \)58-59, and presented below. First number in each row, namely \( l \), is the length of line orbit. It is followed by the the line that is representative of line orbit and others can be obtained by applying \( l-1 \) times given automorphism \( \sigma \). For brevity, designs are not presented with help of automorphism of order 6 (except those with full automorphism group of order 6). Also, common lines for more designs are written only once. We also presented the full automorphism groups of constructed designs.\\ common lines for designs 1-53 ~1:~~~~~ 0~~ 1~~ 2~~ 3~~ 4~ 21~ 22~ 23~ 24~ 25~ 26~ 27~ 28~ 29~ 30~ 31~ 32 ~3:~~~~~ 1~~ 2~~ 3~~ 4~~ 9~ 33~ 36~ 39~ 42~ 45~ 48~ 51~ 54~ 57~ 60~ 63~ 66 ~3:~~~~~12~ 21~ 24~ 27~ 30~ 33~ 34~ 36~ 37~ 41~ 44~ 53~ 56~ 63~ 64~ 66~ 67 ~3:~~~~~15~ 21~ 24~ 27~ 30~ 35~ 38~ 45~ 46~ 48~ 49~ 57~ 58~ 60~ 61~ 65~ 68 ~3:~~~~~18~ 21~ 24~ 27~ 30~ 39~ 40~ 42~ 43~ 47~ 50~ 51~ 52~ 54~ 55~ 59~ 62 ~4:~~~~~ 0~~ 5~~ 9~ 10~ 11~ 21~ 22~ 23~ 33~ 34~ 35~ 39~ 40~ 41~ 45~ 46~ 47 ~4:~~~~~ 1~~ 5~~ 6~~ 7~~ 8~ 21~ 22~ 23~ 36~ 37~ 38~ 51~ 52~ 53~ 60~ 61~ 62 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - design 1 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 34~ 39~ 43~ 48~ 53~ 59~ 61~ 68 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 28~ 40~ 47~ 49~ 53~ 54~ 60~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 24~ 31~ 38~ 40~ 44~ 45~ 58~ 59~ 63~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 29~ 31~ 35~ 37~ 42~ 47~ 49~ 55~ 56~ 63 ~ design 2 12:~ ~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 34~ 40~ 42~ 48~ 53~ 58~ 62~ 68 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 28~ 41~ 45~ 49~ 52~ 54~ 62~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 26~ 31~ 37~ 41~ 43~ 47~ 57~ 58~ 63~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 28~ 32~ 35~ 36~ 43~ 46~ 50~ 54~ 56~ 64 ~ design 3 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 34~ 39~ 43~ 50~ 53~ 58~ 60~ 68 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 28~ 40~ 46~ 48~ 53~ 54~ 62~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 24~ 31~ 38~ 40~ 44~ 47~ 57~ 58~ 63~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 27~ 32~ 33~ 38~ 43~ 47~ 49~ 54~ 56~ 64 ~ design 4 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 34~ 41~ 42~ 48~ 52~ 59~ 61~ 68 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 28~ 39~ 47~ 49~ 52~ 56~ 60~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 25~ 32~ 36~ 40~ 44~ 46~ 57~ 59~ 64~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 29~ 31~ 35~ 37~ 44~ 47~ 49~ 54~ 55~ 63 ~ design 5 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 34~ 39~ 44~ 48~ 52~ 58~ 62~ 68 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 28~ 40~ 45~ 49~ 51~ 56~ 62~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 24~ 32~ 38~ 41~ 43~ 45~ 58~ 59~ 64~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 28~ 32~ 35~ 36~ 42~ 46~ 50~ 55~ 56~ 64 ~ design 6 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 34~ 41~ 42~ 49~ 52~ 57~ 62~ 68 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 28~ 39~ 45~ 50~ 52~ 56~ 61~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 25~ 32~ 36~ 40~ 44~ 47~ 57~ 58~ 64~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 28~ 30~ 34~ 36~ 43~ 47~ 49~ 54~ 56~ 65 ~ design 7 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 34~ 39~ 44~ 49~ 52~ 59~ 60~ 68 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 28~ 40~ 46~ 50~ 51~ 56~ 60~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 24~ 32~ 38~ 41~ 43~ 46~ 57~ 59~ 64~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 27~ 31~ 34~ 38~ 44~ 46~ 50~ 54~ 55~ 63 ~ design 8 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 34~ 41~ 42~ 50~ 52~ 58~ 60~ 68 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 28~ 39~ 46~ 48~ 52~ 56~ 62~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 25~ 32~ 36~ 40~ 44~ 45~ 58~ 59~ 64~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 27~ 32~ 33~ 38~ 42~ 47~ 49~ 55~ 56~ 64 ~ design 9 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 34~ 39~ 44~ 50~ 52~ 57~ 61~ 68 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 28~ 40~ 47~ 48~ 51~ 56~ 61~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 24~ 32~ 38~ 41~ 43~ 47~ 57~ 58~ 64~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 29~ 30~ 33~ 37~ 43~ 46~ 50~ 54~ 56~ 65 ~ design 10 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 40~ 42~ 48~ 53~ 58~ 62~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 39~ 46~ 50~ 53~ 55~ 60~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 26~ 31~ 36~ 41~ 43~ 47~ 57~ 58~ 65~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 28~ 32~ 34~ 38~ 43~ 46~ 50~ 54~ 56~ 63 ~ design 11 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 39~ 43~ 48~ 53~ 59~ 61~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 41~ 45~ 50~ 51~ 55~ 61~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 24~ 31~ 37~ 40~ 44~ 45~ 58~ 59~ 65~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 29~ 31~ 34~ 36~ 42~ 47~ 49~ 55~ 56~ 65 ~ design 12 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 40~ 42~ 49~ 53~ 59~ 60~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 39~ 47~ 48~ 53~ 55~ 61~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 26~ 31~ 36~ 41~ 43~ 45~ 58~ 59~ 65~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 27~ 31~ 33~ 37~ 42~ 46~ 50~ 55~ 56~ 65 ~ design 13 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 39~ 43~ 49~ 53~ 57~ 62~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 41~ 46~ 48~ 51~ 55~ 62~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 24~ 31~ 37~ 40~ 44~ 46~ 57~ 59~ 65~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 28~ 30~ 33~ 38~ 44~ 47~ 49~ 54~ 55~ 64 ~ design 14 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 40~ 42~ 50~ 53~ 57~ 61~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 39~ 45~ 49~ 53~ 55~ 62~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 26~ 31~ 36~ 41~ 43~ 46~ 57~ 59~ 65~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 29~ 30~ 35~ 36~ 44~ 46~ 50~ 54~ 55~ 64 ~ design 15 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 39~ 43~ 50~ 53~ 58~ 60~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 41~ 47~ 49~ 51~ 55~ 60~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 24~ 31~ 37~ 40~ 44~ 47~ 57~ 58~ 65~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 27~ 32~ 35~ 37~ 43~ 47~ 49~ 54~ 56~ 63 ~ design 16 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 41~ 42~ 48~ 52~ 59~ 61~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 40~ 45~ 50~ 53~ 54~ 61~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 25~ 32~ 38~ 40~ 44~ 46~ 57~ 59~ 63~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 29~ 31~ 34~ 36~ 44~ 47~ 49~ 54~ 55~ 65 ~ design 17 12:~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 39~ 44~ 48~ 52~ 58~ 62~ 67 12:~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 41~ 46~ 50~ 52~ 54~ 60~ 65~ 67 12:~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 24~ 32~ 37~ 41~ 43~ 45~ 58~ 59~ 63~ 68 12:~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 28~ 32~ 34~ 38~ 42~ 46~ 50~ 55~ 56~ 63 ~ design 18 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 41~ 42~ 49~ 52~ 57~ 62~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 40~ 46~ 48~ 53~ 54~ 62~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 25~ 32~ 38~ 40~ 44~ 47~ 57~ 58~ 63~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 28~ 30~ 33~ 38~ 43~ 47~ 49~ 54~ 56~ 64 ~ design 19 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 39~ 44~ 49~ 52~ 59~ 60~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 41~ 47~ 48~ 52~ 54~ 61~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 24~ 32~ 37~ 41~ 43~ 46~ 57~ 59~ 63~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 27~ 31~ 33~ 37~ 44~ 46~ 50~ 54~ 55~ 65 ~ design 20 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 41~ 42~ 50~ 52~ 58~ 60~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 40~ 47~ 49~ 53~ 54~ 60~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 25~ 32~ 38~ 40~ 44~ 45~ 58~ 59~ 63~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 27~ 32~ 35~ 37~ 42~ 47~ 49~ 55~ 56~ 63 ~ design 21 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 39~ 44~ 50~ 52~ 57~ 61~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 41~ 45~ 49~ 52~ 54~ 62~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 24~ 32~ 37~ 41~ 43~ 47~ 57~ 58~ 63~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 29~ 30~ 35~ 36~ 43~ 46~ 50~ 54~ 56~ 64 ~ design 22 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 40~ 44~ 48~ 51~ 59~ 61~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 39~ 45~ 50~ 52~ 56~ 61~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 26~ 30~ 36~ 40~ 44~ 47~ 57~ 58~ 64~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 29~ 31~ 34~ 36~ 43~ 47~ 49~ 54~ 56~ 65 ~ design 23 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 41~ 43~ 48~ 51~ 58~ 62~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 40~ 46~ 50~ 51~ 56~ 60~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 25~ 30~ 38~ 41~ 43~ 46~ 57~ 59~ 64~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 28~ 32~ 34~ 38~ 44~ 46~ 50~ 54~ 55~ 63 ~ design 24 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 40~ 44~ 49~ 51~ 57~ 62~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 39~ 46~ 48~ 52~ 56~ 62~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 26~ 30~ 36~ 40~ 44~ 45~ 58~ 59~ 64~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 28~ 30~ 33~ 38~ 42~ 47~ 49~ 55~ 56~ 64 ~ design 25 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 41~ 43~ 49~ 51~ 59~ 60~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 40~ 47~ 48~ 51~ 56~ 61~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 25~ 30~ 38~ 41~ 43~ 47~ 57~ 58~ 64~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 27~ 31~ 33~ 37~ 43~ 46~ 50~ 54~ 56~ 65 ~ design 26 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 40~ 44~ 50~ 51~ 58~ 60~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 39~ 47~ 49~ 52~ 56~ 60~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 26~ 30~ 36~ 40~ 44~ 46~ 57~ 59~ 64~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 27~ 32~ 35~ 37~ 44~ 47~ 49~ 54~ 55~ 63 ~ design 27 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 41~ 43~ 50~ 51~ 57~ 61~ 67 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 29~ 40~ 45~ 49~ 51~ 56~ 62~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 25~ 30~ 38~ 41~ 43~ 45~ 58~ 59~ 64~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 29~ 30~ 35~ 36~ 42~ 46~ 50~ 55~ 56~ 64 ~ design 28 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 34~ 42~ 46~ 48~ 53~ 55~ 62~ 65 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 31~ 39~ 43~ 47~ 53~ 57~ 61~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 29~ 31~ 35~ 36~ 40~ 44~ 49~ 57~ 59~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 26~ 28~ 37~ 41~ 47~ 49~ 54~ 55~ 65~ 66 ~ design 29 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 34~ 42~ 45~ 49~ 52~ 56~ 62~ 65 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 31~ 41~ 43~ 46~ 51~ 57~ 62~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 28~ 32~ 35~ 37~ 41~ 43~ 48~ 58~ 59~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 24~ 28~ 38~ 39~ 46~ 50~ 55~ 56~ 64~ 66 ~ design 30 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 34~ 42~ 45~ 50~ 53~ 55~ 61~ 65 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 31~ 39~ 43~ 46~ 53~ 59~ 60~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 29~ 31~ 35~ 36~ 40~ 44~ 48~ 58~ 59~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 24~ 29~ 38~ 39~ 47~ 49~ 55~ 56~ 63~ 67 ~ design 31 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 34~ 42~ 47~ 48~ 52~ 56~ 61~ 65 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 31~ 41~ 43~ 45~ 51~ 59~ 61~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 28~ 32~ 35~ 37~ 41~ 43~ 50~ 57~ 58~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 25~ 29~ 36~ 40~ 46~ 50~ 54~ 56~ 65~ 67 ~ design 32 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 34~ 42~ 47~ 49~ 53~ 55~ 60~ 65 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 31~ 39~ 43~ 45~ 53~ 58~ 62~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 29~ 31~ 35~ 36~ 40~ 44~ 50~ 57~ 58~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 25~ 27~ 36~ 40~ 47~ 49~ 54~ 56~ 64~ 68 ~ design 33 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 34~ 44~ 45~ 49~ 51~ 55~ 62~ 65 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 31~ 40~ 42~ 46~ 53~ 57~ 62~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 29~ 30~ 33~ 38~ 41~ 43~ 49~ 57~ 59~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 24~ 28~ 38~ 41~ 46~ 50~ 54~ 55~ 64~ 66 ~ design 34 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 34~ 44~ 47~ 48~ 51~ 55~ 61~ 65 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 31~ 40~ 42~ 45~ 53~ 59~ 61~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 29~ 30~ 33~ 38~ 41~ 43~ 48~ 58~ 59~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 25~ 29~ 36~ 39~ 46~ 50~ 55~ 56~ 65~ 67 ~ design 35 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 34~ 44~ 45~ 50~ 52~ 54~ 61~ 65 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 26~ 31~ 41~ 42~ 46~ 52~ 59~ 60~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 27~ 32~ 33~ 37~ 40~ 44~ 49~ 57~ 59~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 24~ 29~ 38~ 41~ 47~ 49~ 54~ 55~ 63~ 67 ~ design 36 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 42~ 45~ 49~ 52~ 56~ 62~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 39~ 44~ 47~ 52~ 58~ 60~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 28~ 32~ 34~ 36~ 41~ 43~ 48~ 58~ 59~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 24~ 28~ 37~ 39~ 46~ 50~ 55~ 56~ 63~ 68 ~ design 37 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 42~ 45~ 49~ 52~ 56~ 62~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 39~ 44~ 47~ 52~ 58~ 60~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 28~ 32~ 34~ 36~ 41~ 43~ 48~ 58~ 59~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 24~ 28~ 37~ 39~ 46~ 50~ 55~ 56~ 63~ 68 ~ design 38 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 42~ 47~ 48~ 52~ 56~ 61~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 39~ 44~ 46~ 52~ 57~ 62~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 28~ 32~ 34~ 36~ 41~ 43~ 50~ 57~ 58~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 25~ 29~ 38~ 40~ 46~ 50~ 54~ 56~ 64~ 66 ~ design 39 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 42~ 45~ 50~ 53~ 55~ 61~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 40~ 44~ 47~ 51~ 57~ 61~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 29~ 31~ 34~ 38~ 40~ 44~ 48~ 58~ 59~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 24~ 29~ 37~ 39~ 47~ 49~ 55~ 56~ 65~ 66 ~ design 40 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 42~ 46~ 50~ 52~ 56~ 60~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 39~ 44~ 45~ 52~ 59~ 61~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 28~ 32~ 34~ 36~ 41~ 43~ 49~ 57~ 59~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 26~ 27~ 36~ 41~ 46~ 50~ 54~ 55~ 65~ 67 ~ design 41 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 42~ 47~ 49~ 53~ 55~ 60~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 40~ 44~ 46~ 51~ 59~ 60~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 21~ 29~ 31~ 34~ 38~ 40~ 44~ 50~ 57~ 58~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 25~ 27~ 38~ 40~ 47~ 49~ 54~ 56~ 63~ 67 ~ design 42 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 43~ 45~ 49~ 53~ 54~ 62~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 40~ 42~ 47~ 53~ 58~ 60~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 27~ 31~ 33~ 38~ 41~ 43~ 50~ 57~ 58~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 24~ 28~ 37~ 40~ 46~ 50~ 54~ 56~ 63~ 68 ~ design 43 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 43~ 46~ 48~ 51~ 56~ 62~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 41~ 42~ 45~ 52~ 58~ 62~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 28~ 30~ 33~ 37~ 40~ 44~ 48~ 58~ 59~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 26~ 28~ 36~ 39~ 47~ 49~ 55~ 56~ 64~ 68 ~ design 44 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 43~ 47~ 48~ 53~ 54~ 61~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 40~ 42~ 46~ 53~ 57~ 62~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 27~ 31~ 33~ 38~ 41~ 43~ 49~ 57~ 59~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 25~ 29~ 38~ 41~ 46~ 50~ 54~ 55~ 64~ 66 ~ design 45 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 43~ 45~ 50~ 51~ 56~ 61~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 41~ 42~ 47~ 52~ 57~ 61~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 28~ 30~ 33~ 37~ 40~ 44~ 50~ 57~ 58~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 24~ 29~ 37~ 40~ 47~ 49~ 54~ 56~ 65~ 66 ~ design 46 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 43~ 46~ 50~ 53~ 54~ 60~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 40~ 42~ 45~ 53~ 59~ 61~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 27~ 31~ 33~ 38~ 41~ 43~ 48~ 58~ 59~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 26~ 27~ 36~ 39~ 46~ 50~ 55~ 56~ 65~ 67 ~ design 47 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 43~ 47~ 49~ 51~ 56~ 60~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 41~ 42~ 46~ 52~ 59~ 60~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 23~ 28~ 30~ 33~ 37~ 40~ 44~ 49~ 57~ 59~ 68 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 25~ 27~ 38~ 41~ 47~ 49~ 54~ 55~ 63~ 67 ~ design 48 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 44~ 46~ 48~ 52~ 54~ 62~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 39~ 43~ 45~ 53~ 58~ 62~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 27~ 32~ 35~ 36~ 40~ 44~ 50~ 57~ 58~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 26~ 28~ 36~ 40~ 47~ 49~ 54~ 56~ 64~ 68 ~ design 49 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 44~ 45~ 49~ 51~ 55~ 62~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 41~ 43~ 47~ 51~ 58~ 60~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 29~ 30~ 35~ 37~ 41~ 43~ 49~ 57~ 59~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 24~ 28~ 37~ 41~ 46~ 50~ 54~ 55~ 63~ 68 ~ design 50 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 44~ 45~ 50~ 52~ 54~ 61~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 39~ 43~ 47~ 53~ 57~ 61~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 27~ 32~ 35~ 36~ 40~ 44~ 49~ 57~ 59~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 24~ 29~ 37~ 41~ 47~ 49~ 54~ 55~ 65~ 66 ~ design 51 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 44~ 47~ 48~ 51~ 55~ 61~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 41~ 43~ 46~ 51~ 57~ 62~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 29~ 30~ 35~ 37~ 41~ 43~ 48~ 58~ 59~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 21~ 25~ 29~ 38~ 39~ 46~ 50~ 55~ 56~ 64~ 66 ~ design 52 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 29~ 31~ 33~ 35~ 44~ 47~ 49~ 52~ 54~ 60~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 39~ 43~ 46~ 53~ 59~ 60~ 64~ 68 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 27~ 32~ 35~ 36~ 40~ 44~ 48~ 58~ 59~ 67 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 23~ 25~ 27~ 38~ 39~ 47~ 49~ 55~ 56~ 63~ 67 ~ design 53 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 35~ 44~ 46~ 50~ 51~ 55~ 60~ 64 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 16~ 24~ 25~ 32~ 41~ 43~ 45~ 51~ 59~ 61~ 65~ 67 12:~~~~~ 1~~ 8~~ 9~ 13~ 18~ 20~ 22~ 29~ 30~ 35~ 37~ 41~ 43~ 50~ 57~ 58~ 66 12:~~~~~ 1~~ 7~~ 9~ 15~ 17~ 19~ 22~ 26~ 27~ 36~ 40~ 46~ 50~ 54~ 56~ 65~ 67 ~ automorphism \( \sigma \) for designs 1-53: \( \sigma =(1,4,2,3)(5,8,6,7)(9,10,11)(12,13,14)(15,16,17)(18,19,20) \) \( (21,31,26,27,22,32,24,28,23,30,25,29)(33,67,38,63,34,68,36,64,35,66,37,65) \) \( (39,55,44,51,40,56,42,52,41,54,43,53)(45,61,50,57,46,62,48,58,47,60,49,59) \) full automorphism group for designs 1-53: \( Z_{12} \) ~ design 54 ~1:~~~~~19~ 22~ 25~ 28~ 31~ 40~ 41~ 43~ 44~ 45~ 48~ 52~ 53~ 55~ 56~ 57~ 60 ~3:~~~~~ 0~~ 1~~ 2~~ 3~~ 4~ 21~ 22~ 23~ 24~ 25~ 26~ 27~ 28~ 29~ 30~ 31~ 32 ~3:~~~~~ 1~~ 2~~ 3~~ 4~ 10~ 34~ 37~ 40~ 43~ 46~ 49~ 52~ 55~ 58~ 61~ 64~ 67 ~3:~~~~~ 1~~ 2~~ 3~~ 4~ 11~ 35~ 38~ 41~ 44~ 47~ 50~ 53~ 56~ 59~ 62~ 65~ 68 ~3:~~~~~13~ 22~ 25~ 28~ 31~ 34~ 35~ 37~ 38~ 39~ 42~ 51~ 54~ 64~ 65~ 67~ 68 ~4:~~~~~ 1~~ 5~~ 6~~ 7~~ 8~ 21~ 22~ 23~ 36~ 37~ 38~ 51~ 52~ 53~ 60~ 61~ 62 ~4:~~~~~ 1~~ 8~ 11~ 12~ 19~ 20~ 23~ 27~ 31~ 34~ 36~ 40~ 42~ 48~ 58~ 59~ 68 12:~~~~~ 0~~ 5~~ 9~ 10~ 11~ 21~ 22~ 23~ 33~ 34~ 35~ 39~ 40~ 41~ 45~ 46~ 47 12:~~~~~ 0~~ 1~~ 6~ 12~ 15~ 18~ 28~ 32~ 33~ 34~ 42~ 46~ 50~ 52~ 56~ 60~ 65 12:~~~~~ 0~~ 2~~ 5~ 13~ 16~ 19~ 27~ 32~ 37~ 38~ 40~ 45~ 50~ 51~ 56~ 58~ 66 12:~~~~~ 2~~ 6~ 10~ 12~ 13~ 17~ 21~ 22~ 29~ 41~ 42~ 48~ 55~ 58~ 62~ 65~ 66 ~ automorphism \( \sigma \) for design 54: \( \sigma =(0,9,14)(1,66,26,4,36,29,2,63,23,3,33,32)(5,7,6,8)(10,16,17)(11,12,20) \) \( (13,15,18)(21,42,35,30,51,68,24,39,38,27,54,65)(22,45,55,31,60,43,25,48,52,28,57,40) \) \( (34,49,62,67,58,50,37,46,59,64,61,47)(41,53,44,56) \) full automorphism group for design 54: \( F_{39}\times Z_{4} \) ~ common lines for designs 55-57 ~1:~~~~~ 0~~ 1~~ 2~~ 3~~ 4~ 21~ 22~ 23~ 24~ 25~ 26~ 27~ 28~ 29~ 30~ 31~ 32 ~2:~~~~~ 0~~ 5~~ 9~ 10~ 11~ 21~ 22~ 23~ 33~ 34~ 35~ 39~ 40~ 41~ 45~ 46~ 47 ~2:~~~~~ 0~~ 7~ 12~ 13~ 14~ 21~ 22~ 23~ 36~ 37~ 38~ 51~ 52~ 53~ 57~ 58~ 59 ~2:~~~~~ 1~~ 5~~ 6~~ 7~~ 8~ 21~ 22~ 23~ 42~ 43~ 44~ 54~ 55~ 56~ 63~ 64~ 65 ~2:~~~~~ 3~~ 5~~ 6~~ 7~~ 8~ 27~ 28~ 29~ 33~ 34~ 35~ 48~ 49~ 50~ 57~ 58~ 59 ~3:~~~~~ 1~~ 2~~ 9~ 27~ 30~ 33~ 36~ 39~ 42~ 45~ 48~ 51~ 52~ 54~ 55~ 57~ 60 ~3:~~~~~ 3~~ 4~ 12~ 21~ 24~ 39~ 42~ 46~ 47~ 49~ 50~ 53~ 56~ 57~ 60~ 63~ 66 ~3:~~~~~15~ 21~ 24~ 27~ 30~ 34~ 35~ 37~ 38~ 46~ 49~ 51~ 54~ 64~ 65~ 67~ 68 ~3:~~~~~18~ 21~ 24~ 27~ 30~ 33~ 36~ 40~ 41~ 43~ 44~ 58~ 59~ 61~ 62~ 63~ 66 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - design 55 ~6:~~~~~ 0~~ 1~~ 3~~ 6~ 12~ 15~ 18~ 29~ 34~ 36~ 39~ 43~ 47~ 52~ 62~ 64~ 68 ~6:~~~~~ 0~~ 7~~ 9~ 15~ 18~ 22~ 29~ 31~ 41~ 48~ 49~ 51~ 56~ 60~ 61~ 65~ 66 ~6:~~~~~ 1~~ 4~~ 8~~ 9~ 16~ 17~ 18~ 21~ 26~ 35~ 37~ 41~ 50~ 52~ 57~ 61~ 64 ~6:~~~~~ 1~~ 5~~ 9~ 12~ 19~ 20~ 23~ 24~ 28~ 37~ 47~ 48~ 54~ 59~ 61~ 66~ 68 ~6:~~~~~ 1~~ 7~~ 9~ 13~ 14~ 17~ 25~ 30~ 32~ 34~ 39~ 43~ 46~ 50~ 59~ 65~ 66 ~6:~~~~~ 3~~ 5~~ 9~ 10~ 14~ 16~ 24~ 28~ 32~ 36~ 40~ 51~ 56~ 57~ 62~ 64~ 65 ~6:~~~~~ 3~~ 8~~ 9~ 12~ 19~ 20~ 22~ 30~ 32~ 35~ 36~ 41~ 43~ 49~ 53~ 55~ 67 ~6:~~~~~ 5~ 12~ 16~ 17~ 18~ 23~ 25~ 27~ 31~ 35~ 42~ 43~ 46~ 48~ 51~ 53~ 62 ~ design 56 ~6:~~~~~ 0~~ 1~~ 3~~ 6~ 12~ 15~ 18~ 29~ 34~ 36~ 41~ 42~ 47~ 52~ 61~ 64~ 68 ~6:~~~~~ 0~~ 7~~ 9~ 15~ 18~ 22~ 29~ 31~ 40~ 48~ 49~ 51~ 56~ 60~ 62~ 65~ 66 ~6:~~~~~ 1~~ 4~~ 8~~ 9~ 16~ 17~ 18~ 21~ 26~ 35~ 37~ 40~ 50~ 52~ 59~ 60~ 64 ~6:~~~~~ 1~~ 5~~ 9~ 12~ 19~ 20~ 21~ 25~ 29~ 38~ 45~ 49~ 55~ 59~ 61~ 66~ 67 ~6:~~~~~ 1~~ 7~~ 9~ 13~ 14~ 17~ 25~ 30~ 32~ 34~ 41~ 42~ 46~ 50~ 58~ 65~ 66 ~6:~~~~~ 3~~ 5~~ 9~ 10~ 14~ 16~ 24~ 28~ 32~ 36~ 39~ 51~ 56~ 59~ 61~ 64~ 65 ~6:~~~~~ 3~~ 8~~ 9~ 12~ 19~ 20~ 23~ 30~ 31~ 33~ 37~ 41~ 43~ 50~ 51~ 56~ 68 ~6:~~~~~ 5~ 12~ 16~ 17~ 18~ 23~ 25~ 27~ 31~ 35~ 42~ 44~ 46~ 48~ 51~ 53~ 61 ~ design 57 ~6:~~~~~ 0~~ 1~~ 3~~ 6~ 12~ 15~ 18~ 29~ 34~ 36~ 40~ 44~ 47~ 52~ 60~ 64~ 68 ~6:~~~~~ 0~~ 7~~ 9~ 15~ 18~ 22~ 29~ 31~ 39~ 48~ 49~ 51~ 56~ 61~ 62~ 65~ 66 ~6:~~~~~ 1~~ 4~~ 8~~ 9~ 16~ 17~ 18~ 21~ 26~ 35~ 37~ 39~ 50~ 52~ 58~ 62~ 64 ~6:~~~~~ 1~~ 5~~ 9~ 12~ 19~ 20~ 22~ 26~ 27~ 36~ 46~ 50~ 56~ 59~ 61~ 67~ 68 ~6:~~~~~ 1~~ 7~~ 9~ 13~ 14~ 17~ 25~ 30~ 32~ 34~ 40~ 44~ 46~ 50~ 57~ 65~ 66 ~6:~~~~~ 3~~ 5~~ 9~ 10~ 14~ 16~ 24~ 28~ 32~ 36~ 41~ 51~ 56~ 58~ 60~ 64~ 65 ~6:~~~~~ 3~~ 8~~ 9~ 12~ 19~ 20~ 21~ 31~ 32~ 34~ 38~ 41~ 43~ 48~ 52~ 54~ 66 ~6:~~~~~ 5~ 12~ 16~ 17~ 18~ 23~ 25~ 27~ 31~ 35~ 43~ 44~ 46~ 48~ 51~ 53~ 60 ~ automorphism \( \sigma \) for designs 55-57: \( \sigma =(1,2)(3,4)(5,6)(7,8)(9,10,11)(12,13,14)(15,16,17)(18,19,20)(21,25,23,24,22,26) \) \( (27,31,29,30,28,32)(33,37,35,36,34,38)(39,43,41,42,40,44)(45,49,47,48,46,50) \) \( (51,55,53,54,52,56)(57,61,59,60,58,62)(63,67,65,66,64,68) \) full automorphism group for designs 55-57: \( Z_{6} \) ~ design 58 ~1:~~~~~ 1~~ 5~~ 6~~ 7~~ 8~ 21~ 22~ 23~ 36~ 37~ 38~ 51~ 52~ 53~ 57~ 58~ 59 ~1:~~~~~ 2~~ 5~~ 6~~ 7~~ 8~ 24~ 25~ 26~ 33~ 34~ 35~ 54~ 55~ 56~ 60~ 61~ 62 ~1:~~~~~ 3~~ 5~~ 6~~ 7~~ 8~ 27~ 28~ 29~ 39~ 40~ 41~ 48~ 49~ 50~ 63~ 64~ 65 ~1:~~~~~ 4~~ 5~~ 6~~ 7~~ 8~ 30~ 31~ 32~ 42~ 43~ 44~ 45~ 46~ 47~ 66~ 67~ 68 13:~~~~~ 0~~ 1~~ 2~~ 3~~ 4~ 21~ 22~ 23~ 24~ 25~ 26~ 27~ 28~ 29~ 30~ 31~ 32 13:~~~~~ 0~~ 5~~ 9~ 10~ 11~ 21~ 22~ 23~ 33~ 34~ 35~ 39~ 40~ 41~ 45~ 46~ 47 13:~~~~~ 0~~ 6~~ 9~ 10~ 11~ 24~ 25~ 26~ 36~ 37~ 38~ 42~ 43~ 44~ 48~ 49~ 50 13:~~~~~ 0~~ 7~ 12~ 13~ 14~ 27~ 28~ 29~ 33~ 34~ 35~ 42~ 43~ 44~ 51~ 52~ 53 13:~~~~~ 0~~ 8~ 12~ 13~ 14~ 30~ 31~ 32~ 36~ 37~ 38~ 39~ 40~ 41~ 54~ 55~ 56 ~ automorphism \( \sigma \) for design 58: \( \sigma =(0,9,20,10,12,19,18,15,16,11,14,17,13)(1,58,23,59,38,22,21,51,52,57,37,53,36) \) \( (2,61,26,62,35,25,24,54,55,60,34,56,33)(3,39,49,40,63,48,50,27,28,41,65,29,64) \) \( (4,42,46,43,66,45,47,30,31,44,68,32,67) \) full automorphism group for design 58: \( F_{39}:Z_{4} \) ~ design 59 ~1:~~~~~ 0~~ 1~~ 2~~ 3~~ 4~ 21~ 22~ 23~ 24~ 25~ 26~ 27~ 28~ 29~ 30~ 31~ 32 ~4:~~~~~ 0~~ 5~~ 9~ 10~ 11~ 21~ 22~ 23~ 33~ 34~ 35~ 39~ 40~ 41~ 45~ 46~ 47 ~4:~~~~~ 1~~ 5~~ 6~~ 7~~ 8~ 21~ 22~ 23~ 36~ 37~ 38~ 51~ 52~ 53~ 57~ 58~ 59 ~6:~~~~~ 1~~ 2~ 15~ 27~ 30~ 33~ 36~ 39~ 42~ 45~ 46~ 48~ 49~ 51~ 54~ 57~ 60 ~6:~~~~~ 9~ 21~ 24~ 29~ 32~ 39~ 41~ 42~ 44~ 52~ 55~ 57~ 58~ 60~ 61~ 63~ 66 12:~~~~~ 0~~ 1~~ 7~~ 9~ 17~ 18~ 26~ 30~ 34~ 41~ 49~ 53~ 54~ 58~ 63~ 67~ 68 12:~~~~~ 1~~ 4~~ 6~~ 9~ 14~ 15~ 16~ 22~ 28~ 33~ 37~ 39~ 44~ 56~ 62~ 63~ 68 12:~~~~~ 1~~ 5~~ 9~ 12~ 14~ 19~ 21~ 26~ 30~ 43~ 45~ 50~ 52~ 56~ 60~ 64~ 65 12:~~~~~ 5~~ 9~ 17~ 19~ 20~ 22~ 27~ 28~ 32~ 33~ 38~ 43~ 49~ 54~ 55~ 59~ 66 ~ automorphism \( \sigma \) for design 59: \( \sigma =(1,4,2,3)(5,8,6,7)(9,13,11,12,10,14)(15,20,17,19,16,18) \) \( (21,30,26,29,22,31,24,27,23,32,25,28)(33,39,38,44,34,40,36,42,35,41,37,43) \) \( (45,55,50,51,46,56,48,52,47,54,49,53)(57,68,62,64,58,66,60,65,59,67,61,63) \) full automorphism group for design 59: \( Z_{12} \)\\ \textbf{Remark} Only designs 54 and 58 admit an automorphism group of order 13. Hence, all constructed designs except them are new. \begin{thebibliography}{1} \bibitem{cy}V. \'{C}epuli\'{c}, On symmetric block designs (40,13,4) with automorphisms of order 5, Discrete Math. 128 (1994) no. 1--3, 45--60. \bibitem{gap}GAP, Lehrstuhl D fuer Mathematik, RWTH Aachen. \bibitem{l}E. Lander, Symmetric Designs: An Algebraic Approach, Cambridge University Press (1983). \bibitem{w}R. A. Mathon and A. Rosa, \( 2-\left( v,k,\lambda \right) \) Designs of Small Order, in The CRC Handbook of Combinatorial Designs, (ed. C. J. Colbourn and J. H. Dinitz), CRC Press (1996), 3-41. \end{thebibliography} \end{document}