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\title{The Mac Lane method of construction and classification of
extensions of cyclic groups by their automorphism groups}
\author{M.Ku\'zma, Institute of Physics\\
 Higher Pedagogical School, Rejtana 16a\\ 35--310 Rzesz\'ow, Poland\\[3mm]
B.Lulek, Institute of Physics\\
A.~Mickiewicz University, Matejki 48/49\\
60--769 Pozna\'n, Poland
}
\date{~}

\begin{document}
\maketitle
\begin{abstract}
The Mac Lane method has been applied to the construction of the second
cohomology group in the extension $C_{12}\times {\rm Aut}\,C_{12}$. The
method simplifies significantly the difficult problem of construction of
nonequivalent extensions and allows to investigate their structure.
\end{abstract}

\section{Introduction}
Extensions of groups have many applications in physics. Space groups
in crystallography being extensions of a three-dimensional
translation group by a point group is a classical example of such
an application. Many other physical phenomena and theorems can be
described in the formalism of extension of groups, too. From this it
follows that it is necessary to search methods of obtaining the
extensions and investigate their structures.

In this paper we present the application of the Mac Lane method [1,2]
to the construction of extensions of finite cyclic groups by a group
of its automorphisms. Physical motivation for the investigation of such
extensions arises from group-theoretic description of properties
of line polymers whose structure is described by a line group [3].
In this case one-dimensional translations form a cyclic group
whereas a group of automorphisms, according to the Weyl recipe [4],
describes the inner symmetry of a system.

The Mac Lane method allows us to obtain all nonequivalent extensions
expressed by a factor system. Calculations were performed for the cyclic
group  $C_{12}$ having four automorphisms forming a group $D_2$.


\section{Nonequivalent extensions of groups}
A group $G$ is an extension of the "passive" group $T$ by an "active"
group $Q$ under a given operator action $\Delta$ if it has a normal subgroup
$T'\lhd G$ isomorphic with $T$ and if the quotient group $G/T'$ is
isomorphic with $Q$.

The groups $G$, $T$, $Q$ form an exact sequence:
\begin{equation}
0 \longrightarrow T \stackrel{\kappa}{\longrightarrow} G
\stackrel{\omega}{\longrightarrow} Q \longrightarrow 1.
\end{equation}
Denoting the elements of the extension $G$ by $\left<t,q\right>$,
$t\in T$, $q\in Q$,
the multiplication rule in the group $G$ has the form:
\begin{equation}
\left<t,q\right>\left<t',q'\right>=\left<t+qt'+m(q,q'),qq'\right>.
\end{equation}
The factors $m(q,q')$ form the so-called factor system, which fully
characterises an extension $G$.

An extension is described for a given operator action $\Delta:Q\rightarrow
{\rm Aut}\,T$ of the active group $Q$ on a passive group $T$:
\begin{equation}
\Delta(q)=\left(\begin{array}{c}t\\qt\end{array}\right)\quad q\in Q,\;t\in T.
\end{equation}

Two extensions $G$ and $G'$ (for the same groups $Q$ and $T$) are equivalent
if there exists an isomorphism $\chi:G\rightarrow G'$ such that the diagram
which represents sequences of these extensions is commutative i.e.\
$\chi\circ\kappa=\kappa'$ and $\omega'\circ\chi=\omega$
\begin{equation}
\begin{array}{cclclclcc}
0 & \longrightarrow & T  & \stackrel{\kappa}{\longrightarrow}  & G  & \stackrel{
\omega}{\longrightarrow}  & Q & \longrightarrow


& 1\\
  &                 & \downarrow\; {\rm id}\,T &               & \downarrow\;
  \chi &                      & \downarrow\; {\rm
id}\,Q & & \\
0 & \longrightarrow & T' & \stackrel{\kappa'}{\longrightarrow} & G' & \stackrel{
\omega'}{\longrightarrow} & Q & \longrightarrow


& 1
\end{array}
\end{equation}
The equivalency of extensions can also be defined basing on a factor system.
Namely two extensions are equivalent if their factor systems differ from
each other by a twocoboundary $\delta c$
\begin{equation}
m' = m + \delta c.
\end{equation}


\section{Second group of cohomology}

For given groups $Q$ and $T$ one can obtain many extensions.  But not all of
them differ. Some of them are equivalent.  The number of nonequivalent
extensions is given by the second cohomology group $H_{\Delta}^2(Q,T)$ [5,6]:
\begin{equation}
H_{\Delta}^2(Q,T)=Z_{\Delta}^2(Q,T)/B_{\Delta}^2(Q,T),
\end{equation}
where $Z_{\Delta}^2(Q,T)$ is the group of all twococycles, while
$B_{\Delta}^2(Q,T)$ is the group of all twocoboundaries. Groups
$Z_{\Delta}^2(Q,T)$ and $B_{\Delta}^2(Q,T)$ are subgroups of all twocochains
$C_{\Delta}^2(Q,T)$. The order of this group is great:
\begin{equation}
|C_{\Delta}^2(Q,T)|=|T|^{(|Q|^2)}.
\end{equation}
This order increases  when the orders of the groups $T$ and $Q$ are increased
(combinatorial explosion) and e.g.\ for such small groups $Q$ and $T$
as $|Q|=3$, $|T|=4$ the order of the group of twococycles is equal to 262144.


\section{The Mac Lane method}

Mac Lane's theorem helps to construct the second cohomology
group and enables us to inspect some features of the structure of extension.
The essence of the method is  in a theorem that a second
cohomology group is isomorphic with a quotient group for
another exact sequence involving free groups [1]:
\begin{equation}
\parbox{78mm}{
\setlength{\unitlength}{1mm}
\begin{picture}(75,25)
\put(0,2){\mbox{$0$}}
\put(0,22){\mbox{$1$}}
\put(20,2){\mbox{$T$}}
\put(20,22){\mbox{$R$}}
\put(40,2){\mbox{$G$}}
\put(40,22){\mbox{$F$}}
\put(50,12){\mbox{$Q$}}
\put(70,12){\mbox{$1$}}
\put(7,3.2){\vector(1,0){12}}
\put(7,23.2){\vector(1,0){12}}
\put(25,3.2){\vector(1,0){14}}
\put(25,23.2){\vector(1,0){14}}
\put(44,3.2){\vector(1,1){7}}
\put(44,23.2){\vector(1,-1){7}}
\put(55,13.2){\vector(1,0){13.5}}
\put(22,20){\vector(0,-1){13}}
\put(40,20){\vector(-1,-1){13}}
\put(32.5,4){\mbox{$\kappa$}}
\put(32.5,24){\mbox{$i$}}
\put(45,9){\mbox{$\omega$}}
\put(49,19){\mbox{$M$}}
\put(29,13.5){\mbox{$\gamma$}}
\put(18,13.5){\mbox{$\varphi$}}
\end{picture}}
\end{equation}
where $F$, $R$ are free groups with alphabet $\left<X\right>$ and
$\left<Y\right>$ respectively.

The group $F$ is generated freely from a set $A\subset Q$ of generators of an
active group $Q$. The free group $R\lhd F$ is a quotient group of $F$ and
its alphabet is formed using the Nielson--Schreier theorem [5]:
\begin{equation}
Y=\{ sx\beta(sx)^{-1}\mid x\in X,\;s\in S\},\quad S=\{f_q\mid q\in Q\},
\end{equation}
where $S$ is the Nielson--Schreier set, $f_q$ --- representatives of cosets of
$F$.  In diagram (8) $\varphi$ denotes an operator homomorphisms from $R$ to $T$
and $\gamma$ denotes a crossed homomorphisms from $F$ to $T$.

Denoting the set of all operator and crossed homomorphisms by $Z^1_{\Delta\circ
M}(F,T)$ we can derive the second cohomology group from an isomorphism:
\begin{equation}
H^2_{\Delta}(Q,T)\cong {\rm Hom}_F(R,T)/Z^1_{\Delta\circ M}(F,T)|_R,
\end{equation}
where $Z^1_{\Delta\circ M}(F,T)$ is restricted to $R$.

One can construct a group ${\rm Hom}_F\,(R,T)$ from a manifold of all mappings
from the alphabet $Y$ of the group $R$ on to group $T$
\begin{equation}
{\rm Hom}_F\,(R,T)=T^Y=\{\varphi\mid Y\rightarrow T\}.
\end{equation}
Then we have to select from this manifold $T^Y$ the submanifold
of all operator homomorphisms i.e.\ such mappings, which intertwine
the action $\Xi:F\rightarrow{\rm Aut}\,R$ and $\Delta:Q\rightarrow{\rm Aut}\,
T$. They form conditions for the operator
homomorphism:
\begin{equation}
\varphi(xyx^{-1})=M(x)\varphi(y)\quad x\in X,\; y\in Y.
\end{equation}
The group $Z^1_{\Delta\circ M}(F,T)$ of crossed homomorphisms is derived from
other rules:
\begin{equation}
\gamma(f_1f_2)=\gamma(f_1)+M(f_1)\gamma(f_2),\quad f_1f_2\in F^2.
\end{equation}

\section{Extensions of $C_{12}\times{\rm Aut}\,C_{12}$}
The presented method has been applied to the construction of all
nonequivalent extensions of groups $C_{12}\times{\rm Aut}\,C_{12}$, where
the translation group $T$ is a cyclic group $C_{12}=\{j\mid j=1,2,\ldots,12\}$
and the point group $Q$ is the group of automorphisms ${\rm
Aut}\,C_{12}=\{\tau_r\mid r=1,5,7,11\}$. The latter one has four elements
$\tau_r$ described by relation $\tau_rj=rj\bmod12$, $j\in C_{12}$. This group
(cf also Table 1) is isomorphic to the group $D_2$. All possible operator
actions $\Delta: Q \rightarrow {\rm Aut}\,C_{12}$ have been listed in Table 2.

\begin{table}
\caption{The group ${\rm Aut}\,C_{12}$}
\begin{center}
\begin{tabular}{|c|cccccccccccc|}\hline
$\tau \backslash j$ &1 &2 &3 &4 &5 &6 &7 &8 &9 &10 &11 & 12 \\ \hline
$\tau_1$ & 1&2 &3 &4 &5 &6 &7 &8 &9 &10&11&12 \\
$\tau_5$ & 5&10&3 &8 &1 &6 &11&4 &9 &2 &7 &12 \\
$\tau_7$ & 7&2 &9 &4 &11&6 &1 &8 &3 &10&5 &12 \\
$\tau_{11}$ & 11&10&9 &8 &7 &6 &5 &4 &3 &2 &1 &12 \\ \hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption{Operator actions $\Delta: Q \rightarrow{\rm Aut}\,C_{12}$}
\begin{center}
\renewcommand{\tabcolsep}{1mm}
\begin{tabular}{|c|cccccccccccccccc|} \hline
$\tau\backslash\Delta $&$\Delta_{1}$&$\Delta_{2}$ &$\Delta_{3}$
&$\Delta_{4}$
&$\Delta_{5}$&$\Delta_{6}$&$\Delta_{7}$&$\Delta_{8}$&$\Delta_{9}$
&$\Delta_{10}$&$\Delta_{11}$
&$\Delta_{12}$&$\Delta_{13}$&$\Delta_{14}$&$\Delta_{15}$
&$\Delta_{16}$\\ \hline
$ \tau_1  $&$\tau_{1 } $&$\tau_{1} $ &$\tau_{1} $ &$\tau_{1}
$&$\tau_{1}$ &$\tau_{1}$ &$\tau_{1} $ &$\tau_{1} $ &$\tau_{1}$
&$\tau_{1} $ &$\tau_{1} $ &$\tau_{1} $ &$\tau_{1} $ &$\tau_{1} $
&$\tau_{1} $ &$\tau_{1} $\\
$ \tau_5 $&$\tau_{1 } $&$\tau_{1} $
&$\tau_{1} $ &$\tau_{1} $&$\tau_{5}$ &$\tau_{5}$ &$\tau_{5} $
&$\tau_{5} $ &$\tau_{7}$ &$\tau_{7} $ &$\tau_{7} $ &$\tau_{7} $
&$\tau_{11} $ &$\tau_{11}$ &$\tau_{11} $ &$\tau_{11} $\\
$ \tau_7  $&$\tau_{1 } $&$\tau_{5} $
&$\tau_{7} $ &$\tau_{11}$&$\tau_{1}$ &$\tau_{5}$ &$\tau_{7} $
&$\tau_{11} $ &$\tau_{1}$ &$\tau_{7}
 $ &$\tau_{5} $ &$\tau_{11}$ &$\tau_{1} $ &$\tau_{5} $ &$\tau_{7} $
&$\tau_{11} $\\
$ \tau_{11}  $&$\tau_{1 } $&$\tau_{5} $ &$\tau_{7} $
&$\tau_{11}$&$\tau_{5}$ &$\tau_{1}$ &$\tau_{11}$ &$\tau_{7} $ &$\tau_{7}$
&$\tau_{1} $ &$\tau_{11}$ &$\tau_{5} $ &$\tau_{11} $ &$\tau_{7} $ &$\tau_{5}
$ &$\tau_{1} $\\
\hline
\end{tabular}
\end{center}
\end{table}

We express nonequivalent extensions for the operator action $\Delta_7$ (Table
2) in terms of a factor system defined by
\begin{equation}
m(q_1,q_2)=\varphi(\varrho(q_1,q_2)),\quad (q_1,q_2)\in Q^2.
\end{equation}
The function $\varrho(q_1,q_2)$ in (14) is described by a product
\begin{equation}
f_{q_1}f_{q_2}=\varrho(q_1,q_2)f_{q_1q_2},
\end{equation}
where $f_q$ is a representative of the coset in the decomposition
\begin{equation}
F=\bigcup_{q\in Q}Rf_q,
\end{equation}
and $R$ is the kernel of the epimorphism $M:F\rightarrow Q$.
The group $D_2$ has an alphabet $X=\{x_1,x_2\}$ and the Schreier set consists
of four elements $S=\{e_F,x_1,x_2,x_1x_2\}$. This set determines the factor
system $\varrho:Q\times Q\rightarrow R$ by (15). For our case the
factor system $\varrho(q_1,q_2)$ has been presented in Table 3.

\begin{table}
\caption{The factor system $\varrho:Q\times Q\rightarrow R$ in the alphabet $X$}
\begin{center}
\begin{tabular}{|c|cccc|} \hline
  & $e_F$ & $x_1$ & $x_2$ & $x_1x_2$ \\ \hline
$e_F$  & $e_F$ & $e_F$ & $e_F$ & $e_F$ \\
$x_1$  & $e_F$ & $x_1^2$ & $e_F$ & $x_1^2$ \\
$x_2$  & $e_F$ & $x_2x_1x_2^{-1}x_1^{-1}$ & $x_2^2$ & $x_2x_1x_2x_1^{-1}$ \\
$x_1x_2$  & $e_F$ & $x_1x_2x_1x_2^{-1}$ & $x_1x_2^2x_1^{-1}$ & $x_1x_2x_1x_2$ \\
\hline
\end{tabular}
\end{center}
\end{table}

The alphabet $Y$ of the subgroup $R$ can be identify with the set of all
non-trivial elements of the second and third columns of Table 3. Thus we have
\begin{equation}
Y=\{y_1=x_1^2,y_2=x_2^2, y_3=x_2x_1x_2^{-1}x_1^{-1},y_4=x_1x_2x_1x_2^{-1}, y_5=x
_1x_2^2x_1^{-1}.
\end{equation}
The factor system $\varrho:Q\times Q\rightarrow R$, expressed in terms of the
alphabet $Y$ is presented in Table 4.
Having the alphabets $X$ and $Y$ we can construct conditions for operator
homomorphisms (12) (see Table 5). Both, operator and crossed homomorphisms,
forming groups ${\rm Hom}_F\,(R,C_{12})$ and $Z^1_{\Delta\circ M}$, are
collected in Table 6 and 7, respectively.




\begin{table}
\caption{The factor system $\varrho:Q\times Q\rightarrow R$ in the alphabet $Y$}
\begin{center}
\begin{tabular}{|c|cccc|} \hline
  & $E$ & $u_x$ & $u_y$ & $u_z$ \\ \hline
$E$  & $e_F$ & $e_F$ & $e_F$ & $e_F$ \\
$u_x$  & $e_F$ & $y_1$ & $e_F$ & $y_1$ \\
$u_y$  & $e_F$ & $y_3$ & $y_2$ & $y_3y_5$ \\
$u_z$  & $e_F$ & $y_4$ & $y_5$ & $y_4y_2$ \\
\hline
\end{tabular}
\end{center}
\end{table}


\begin{table}
\caption{Conditions for operator homomorphisms $\varphi:R\rightarrow T$}
\begin{center}
\begin{tabular}{|c|cc|} \hline
  & $x_1$ & $x_2$ \\ \hline
$y_1$  & $\varphi(y_1)=5\varphi(y_1)$ & $\varphi(y_3)+\varphi(y_4)=7\varphi(y_1)
$ \\
$y_2$  & $\varphi(y_5)=5\varphi(y_2)$ & $\varphi(y_2)=7\varphi(y_2)$ \\
$y_3$  & $\varphi(y_4)-\varphi(y_1)=5\varphi(y_3)$ & $\varphi(y_2)-\varphi(y_5)-
\varphi(y_3)=7\varphi(y_3)$ \\
$y_4$  & $\varphi(y_1)+\varphi(y_3)=5\varphi(y_4)$ & $\varphi(y_1)-\varphi(y_2)+
\varphi(y_3)+\varphi(y_5)=7\varphi(y_4)$ \\
$y_5$  & $\varphi(y_2)=5\varphi(y_5)$ & $\varphi(y_5)=7\varphi(y_5)$ \\
\hline
\end{tabular}
\end{center}
\end{table}


\begin{table}
\caption{The group ${\rm Hom}_F\,(R,C_{12})$ of operator homomorphisms (for
operation action $\Delta_7$)}
\begin{center}
\begin{tabular}{|c|ccccc||c|ccccc|} \hline
  & $y_1$ & $y_2$ & $y_3$ & $y_4$  & $y_5$ &
  & $y_1$ & $y_2$ & $y_3$ & $y_4$ & $y_5$  \\ \hline
$\varphi_1$ & 3 & 2 & 5 & 4 & 10 & $\varphi_{25}$ & 9 & 2 & 5 & 10 & 10\\
$\varphi_2$ & 3 & 2 & 11& 10& 10 & $\varphi_{26}$ & 9 & 2 & 11&  4 & 10\\
$\varphi_3$ & 3 & 4 & 1 & 8 &  8 & $\varphi_{27}$ & 9 & 4 & 1 &  2 &  8\\
$\varphi_4$ & 3 & 4 & 7 & 2 &  8 & $\varphi_{28}$ & 9 & 4 & 7 &  8 &  8\\
$\varphi_5$ & 3 & 6 & 3 & 6 &  6 & $\varphi_{29}$ & 9 & 6 & 3 & 12 &  6\\
$\varphi_6$ & 3 & 6 & 9 & 12&  6 & $\varphi_{30}$ & 9 & 6 & 9 &  6 &  6\\
$\varphi_7$ & 3 & 8 & 5 & 4 &  4 & $\varphi_{31}$ & 9 & 8 & 5 & 10 &  4\\
$\varphi_8$ & 3 & 8 & 11& 10&  4 & $\varphi_{32}$ & 9 & 8 &11 &  4 &  4\\
$\varphi_9$ & 3 & 10& 1 & 8 &  2 & $\varphi_{33}$ & 9 & 10& 1 &  2 &  2\\
$\varphi_{10}$ & 3 & 10& 7 & 2 &  2 & $\varphi_{34}$ & 9 & 10& 7 &  8 &  2\\
$\varphi_{11}$ & 3 & 12& 3 & 6 & 12 & $\varphi_{35}$ & 9 & 12& 3 & 12 & 12\\
$\varphi_{12}$ & 3 &12 & 9 & 12& 12 & $\varphi_{36}$ & 9 &12 & 9 &  6 & 12\\
$\varphi_{13}$ & 6 & 2 & 2 & 4 & 10 & $\varphi_{37}$ &12 & 2 & 2 & 10 & 10\\
$\varphi_{14}$ & 6 & 2 & 8 &10 & 10 & $\varphi_{38}$ &12 & 2 & 8 &  4 & 10\\
$\varphi_{15}$ & 6 & 4 & 4 & 2 & 8  & $\varphi_{39}$ &12 & 4 & 4 &  8 &  8\\
$\varphi_{16}$ & 6 & 4 & 10& 8 &  8 & $\varphi_{40}$ &12 & 4 & 10&  2 &  8\\
$\varphi_{17}$ & 6 & 6 & 6 & 12&  6 & $\varphi_{41}$ &12 & 6 & 6 &  6 &  6\\
$\varphi_{18}$ & 6 & 6 &12 & 6 &  6 & $\varphi_{42}$ &12 & 6 & 12& 12 &  6\\
$\varphi_{19}$ & 6 & 8 & 2 & 4 &  4 & $\varphi_{43}$ &12 & 8 & 2 & 10 &  4\\
$\varphi_{20}$ & 6 & 8 & 8 & 10&  4 & $\varphi_{44}$ &12 & 8 & 8 &  4 &  4\\
$\varphi_{21}$ & 6 & 10& 4 & 2 &  2 & $\varphi_{45}$ &12 &10 & 4 &  8 &  2\\
$\varphi_{22}$ & 6 & 10&10 & 8 &  2 & $\varphi_{46}$ &12 &10 & 10&  2 &  2\\
$\varphi_{23}$ & 6 & 12& 6 & 12& 12 & $\varphi_{47}$ &12 &12 & 6 &  6 & 12\\
$\varphi_{24}$ & 6 & 12&12 & 6 & 12 & $\varphi_{48}$ &12 &12 & 12& 12 & 12\\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption{The group $Z^1_{\Delta\circ M}$ of crossed homomorphisms (for
operation action $\Delta_7$)}
\begin{center}
\begin{tabular}{|c|ccccc|} \hline
  & $y_1$ & $y_2$ & $y_3$ & $y_4$  & $y_5$ \\ \hline
$\gamma_1$ & 6 & 8 & 2 & 4 &  4 \\
$\gamma_2$ & 6 & 4 & 10& 8 &  8 \\
$\gamma_3$ & 6 & 12& 6 & 12& 12 \\
$\gamma_4$ &12 & 8 & 8 & 4 &  4 \\
$\gamma_5$ &12 & 4 & 4 & 8 &  8 \\
$\gamma_6$ &12 & 12& 12& 12& 12 \\
\hline
\end{tabular}
\end{center}
\end{table}

According to formula (6) we construct cosets determined by $Z^1_{\Delta\circ
M}(F,T)$ in ${\rm Hom}_F\,(R,T)$ (see Table 8) forming the second cohomology
group $H^2_{\Delta_7}(Q,T)$. This group has eight elements. It yields that we
have eight nonequivalent extensions $C_{12}\times{\rm Aut}\,C_{12}$ (under
operator action $\Delta_7$). Such extensions are listed in Table 9 in the form
of factor systems $m:Q\times Q\rightarrow C_{12}$ for the coset
representatives chosen as in Table 8.


\begin{table}
\caption{Coset representatives ${\rm Hom}_F\,(R,T)/Z^1_{\Delta\circ M}(F,T)$
 (for operation action $\Delta_7$)}
\begin{center}
\begin{tabular}{|c|ccccc|} \hline
  & $y_1$ & $y_2$ & $y_3$ & $y_4$  & $y_5$ \\ \hline
$r_0$ &12 & 12& 12& 12& 12 \\
$r_1$ & 9 & 6 & 9 & 6 &  6 \\
$r_2$ & 3 & 6 & 9 & 12&  6 \\
$r_3$ & 3 & 12& 3 &  6& 12 \\
$r_4$ & 9 & 12& 3 & 12& 12 \\
$r_5$ & 6 &  6& 12& 6 &  6 \\
$r_6$ & 6 &  6&  6& 12&  6 \\
$r_7$ & 6 & 12& 12& 6 &  12\\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption{Factor systems $m:Q\times Q\rightarrow C_{12}$
 (for operation action $\Delta_7$)}
\begin{center}
\begin{tabular}{cccc}
$r_0=$ & $\left(\begin{array}{rrrr}
                0 & 0 & 0 & 0 \\
                0 & 0 & 0 & 0 \\
                0 & 0 & 0 & 0 \\
                0 & 0 & 0 & 0 \\
         \end{array}\right)$ &
$r_1=$ & $\left(\begin{array}{rrrr}
                0 & 0 & 0 & 0 \\
                0 & 9 & 6 & 3 \\
                0 & 9 & 6 & 3 \\
                0 & 6 & 6 & 0 \\
         \end{array}\right)$ \\[5mm]
$r_2=$ & $\left(\begin{array}{rrrr}
                0 & 0 & 0 & 0 \\
                0 & 3 & 0 & 3 \\
                0 & 9 & 6 & 3 \\
                0 & 0 & 6 & 6 \\
         \end{array}\right)$ &
$r_3=$ & $\left(\begin{array}{rrrr}
                0 & 0 & 0 & 0 \\
                0 & 3 & 0 & 3 \\
                0 & 3 & 0 & 3 \\
                0 & 6 & 0 & 6 \\
         \end{array}\right)$ \\[5mm]
$r_4=$ & $\left(\begin{array}{rrrr}
                0 & 0 & 0 & 0 \\
                0 & 9 & 0 & 9 \\
                0 & 3 & 0 & 3 \\
                0 & 0 & 0 & 0 \\
         \end{array}\right)$ &
$r_5=$ & $\left(\begin{array}{rrrr}
                0 & 0 & 0 & 0 \\
                0 & 6 & 0 & 6 \\
                0 & 0 & 6 & 6 \\
                0 & 6 & 6 & 0 \\
         \end{array}\right)$ \\[5mm]
$r_6=$ & $\left(\begin{array}{rrrr}
                0 & 0 & 0 & 0 \\
                0 & 6 & 0 & 6 \\
                0 & 6 & 6 & 0 \\
                0 & 0 & 6 & 6 \\
         \end{array}\right)$ &
$r_7=$ & $\left(\begin{array}{rrrr}
                0 & 0 & 0 & 0 \\
                0 & 6 & 0 & 6 \\
                0 & 0 & 0 & 0 \\
                0 & 6 & 0 & 6 \\
         \end{array}\right)$ \\[5mm]
\end{tabular}
\end{center}
\end{table}


\section{Conclusions}
The second cohomology group for an extension $C_{12}\times {\rm Aut}\,C_{12}$
is of the order 8 (under the operator action $\Delta_7$ described in Table 2).
Each element of this group $m\in H^2(D_2,C_{12})$ forms the factor system
(Table 9) determined by the Seitz formula (2) and gives an nonequivalent
extension. However, the factor system depends on a choice of coset
representatives listed in table 8. This choice corresponds to the gauge
transformation [2], which is connected with equivalent extensions.

\newpage
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\end{thebibliography}
\end{document}

