Representations of Group Algebras of Non-Abelian Groups of Orders p3, for a Prime p ≥ 3

In this paper, semidirect products are used to find the matrix representations of group algebras of non-abelian groups of order p³, for a prime p ≥ 3.

Preliminaries

Let $G$ be a group, assume that $H$ is a normal subgroup of $G$, $K$ is a subgroup of $G$, $H\cap K=\left\{1\right\}$, and $G=HK$. Suppose that $K$ acts on $H$ by automorphisms of $H$, then there exists a homomorphism $\phi :K\to Aut(H)$. Assume the action is by conjugation, then for $k\in K$ and $h\in H$ we have $k.h=\varnothing \left(k\right)\left(h\right)=kh{k}^{-1}$. $G$ is an internal semidirect product of $H$ and $K$ by $\phi$, it is denoted by $G=H{⋊}_{\phi }K$ [1].

Non-abelian groups of orders $p^3$, for a prime $p\ge 3$ are of two types [1]:

$$G_1=C_{p^2}⟨\alpha ⟩{⋊}_{\phi }{C}_p⟨\beta ⟩\text{and}{G}_2=(C_p⟨\alpha ⟩\times C_p⟨\beta ⟩){⋊}_{\phi }{C}_p⟨\gamma ⟩.$$

Thus,

$$G_1=⟨\alpha ,\beta :{\alpha }^{p^2}={\beta }^p=1,\beta \alpha ={\alpha }^{1+p}\beta ⟩$$

and

$$G_2=⟨\alpha ,\beta ,\gamma :{\alpha }^p={\beta }^p={\gamma }^p=1,\alpha \beta =\beta \alpha ,\gamma \beta =\beta \gamma ,\gamma \alpha =\alpha \beta \gamma ⟩$$

Let $F$ be a field. A ring $A$ with unity is an algebra over $F$ (breifly $F$-algebra) if $A$ is a vector space over $F$ and the following compatibility condition holds $(sa).b=s(a.b)=a.(sb)$ for any $a,b\in A$ and any $s\in F$. $A$ is also called associative algebra (over $F$). The dimension of the algebra $A$ is the dimension of $A$ as a vector space over $F$.

Theorem 1 [2]

Let $A$ be a $n$-dimensional algebra over a field $F$. Then there is a one-to-one algebra homomorphism from $A$ into $M_n(F)$, the algebra of $n$-matrices over $F$.

Let $G=\{g_1=1,g_2,\dots ,g_n\}$ be a finite group of order $n$ and $F$ a field. Define $FG=\{a_1{g}_1+a_2{g}_2+\dots +a_n{g}_n:a_i\in F\}$. $FG$ is $n$-dimensional vector space over $F$ with basis $G$. Multiplication of $G$ can be extended linearly to $FG$. Thus, $FG$ becomes an algebra over $F$ of dimension $n$. $FG$ is called group algebra. The following identifications should be realized:

1. $0_F{g}_G=0_{FG}=0$ for any $g\in G$.
2. $1_F{g}_G=g_{FG}=g$ for any $g\in G$. In particular $1_F{1}_G=1_{FG}=1$.
3. $a_F{1}_G=a_{FG}$ for any $a\in F$.

A circulant matrix $M$ on parameters $a_0,a_1,\dots ,a_{n-1}$ is defined as follows:

$$\begin{array}{r}M(a_0,a_1,\dots ,a_{n-1})=\left[\begin{array}{c}\begin{array}{cc}{a}_0& a_{n-1}\\ a_1& a_0\end{array}\begin{array}{cc}\cdots & a_1\\ \cdots & a_2\end{array}\\ \begin{array}{cc}⋮& ⋮\\ a_{n-1}& a_{n-2}\end{array}\begin{array}{cc}& ⋮\\ \cdots & a_0\end{array}\end{array}\right]\end{array}$$

This matrix may be denoted in terms of its columns by $[col\left(a_0\right)\left|col\left(a_{n-1}\right)\right|\dots |col(a_1)]$.

$M$ is said to be circulant block matrix if it is of the form $M(M_1,M_2,\dots ,M_n).$ i.e., it is circulant blockwise on the blocks $M_1,M_2,\dots ,M_n$.

Thus,

$$\begin{array}{r}M=\left[\begin{array}{c}\begin{array}{cc}{M}_1& M_n\\ M_2& M_1\end{array}\begin{array}{cc}\cdots & M_2\\ \cdots & M_3\end{array}\\ \begin{array}{cc}⋮& ⋮\\ M_n& M_{n-1}\end{array}\begin{array}{cc}& ⋮\\ \cdots & M_1\end{array}\end{array}\right].\end{array}$$

Main Results

Theorem 2 [3]

Let $F$ be a field and $G=⟨\alpha :{\alpha }^n=1⟩$ a cyclic group of order $n$. Then any element $a_{0}1+a_1\alpha +\dots +a_{n-1}{\alpha }^{n-1}$ of $FG$ can be represented with respect to the ordered basis $\{1,\alpha ,\dots ,{\alpha }^{n-1}\}$ by the circulant matrix $M(a_0,a_1,\dots ,a_{n-1})$.

Proof

Let $w=a_{0}1+a_1\alpha +\dots +a_{n-1}{\alpha }^{n-1}$ be in $FG$. $w\alpha =a_0\alpha +a_1{\alpha }^2+\dots +a_{n-1}1=a_{n-1}1+a_0\alpha +\dots +a_{n-2}{\alpha }^{n-1}\dots w{\alpha }^{n-1}=a_0{\alpha }^{n-1}+a_{1}1+\dots +a_{n-1}{\alpha }^{n-2}=a_{1}1+a_2\alpha +\dots +a_0{\alpha }^{n-1}.$ Then the matrix representation of $w$ with respect to the basis $\{1,\alpha ,\dots ,{\alpha }^{n-1}\}$ is $\left[\begin{array}{c}\begin{array}{cc}{a}_0& a_{n-1}\\ a_1& a_0\end{array}\begin{array}{cc}\cdots & a_1\\ \cdots & a_2\end{array}\\ \begin{array}{cc}⋮& ⋮\\ a_{n-1}& a_{n-2}\end{array}\begin{array}{cc}& ⋮\\ \cdots & a_0\end{array}\end{array}\right]$ which is $M(a_0,a_1,\dots ,a_{n-1})$.

Note that if the order of the basis elements is changed, we obtain a different matrix of representation. The new matrix is obtained by suitable interchanging of the columns of the matrix $M(a_0,a_1,\dots ,a_{n-1})$. In [4] the representation is done for the non-split metacyclic group.

For more complicated finite groups we use the circulant block matrices to do the required representations.

Now, let $G$ be an internal semidirect product of $H$ and a cyclic group $K=⟨\gamma ⟩$ by $\phi$.

Then the matrix representation $\left[w\right]$ of the general element $w$ in $FG$ is given as follows:

$G=H{⋊}_{\phi }K$, $\phi :K\to Aut(H)$ is a homomorphism, $\phi \left(\gamma \right)\left(h\right)=\gamma h{\gamma }^{-1}$. Suppose that $H=\{h_1,h_2,\dots ,h_n\}$, $K=C_m⟨\gamma ⟩=\{1,\gamma ,\dots ,{\gamma }^{m-1}\}$ then the general element $w$ in $FG$ is $w=a_1{h}_{1}1+a_2{h}_{2}1+\dots +a_n{h}_{n}1+a_{n+1}{h}_1\gamma +a_{n+2}{h}_2\gamma +\dots +a_{2n}{h}_n\gamma +a_{2n+1}{h}_1{\gamma }^2+\dots +a_{3n}{h}_n{\gamma }^2+\dots +a_{mn}{h}_n{\gamma }^{m-1}$. Now we can write $w$ as: where

$$\begin{array}{r}w=w_1+w_2+\dots +w_m,\end{array}$$

$$\begin{array}{r}{w}_1=a_1{h}_{1}1+a_2{h}_{2}1+\dots +a_n{h}_{n}1\end{array}$$

$$\begin{array}{r}{w}_2=a_{n+1}{h}_1\gamma +a_{n+2}{h}_2\gamma +\dots +a_{2n}{h}_n\gamma \end{array}$$

$$\begin{array}{r}⋮\end{array}$$

$$\begin{array}{r}{w}_m=a_{(m-1)(n+1)}{h}_1{\gamma }^{m-1}+\dots +a_{mn}{h}_n{\gamma }^{m-1}\end{array}$$

The matrix representation $\left[w\right]$ of $w$ is $\left[w\right]=M(\left[w_1\right],{\left[w_2\right]}^{\gamma },\dots ,{\left[w_m\right]}^{{\gamma }^{m-1}})$, where ${\gamma }^i:H\to H$ is the automorphism ${\gamma }^i=$ $\phi \left(\gamma \right)\left(h\right)={\gamma }^{i}h{\gamma }^{-i}$ and $\left[w_i\right]=[col\left(h_1\right)\left|col\left(h_2\right)\right|\dots |col(h_n)]$, ${\left[w_i\right]}^{{\gamma }^i}=[col({\gamma }^i(h_1))\left|col\left({\gamma }^i\left(h_2\right)\right)\right|\dots \left|col\left({\gamma }^i\left(h_n\right)\right)\right|]$. Thus, we get the following theorem:

Theorem 3

With the above notations, the matrix representation $\left[w\right]$ of the general element $w$ in $FG$.

$$\begin{array}{r}\left[w\right]=\left[\begin{array}{c}\begin{array}{cccc}\left[w_1\right]& {\left[w_m\right]}^{{\gamma }^{m-1}}& \dots & {\left[w_2\right]}^{\gamma }\\ {\left[w_2\right]}^{\gamma }& \left[w_1\right]& \dots & {\left[w_m\right]}^{{\gamma }^2}\\ ⋮& ⋮& ⋮& ⋮\\ {\left[w_m\right]}^{{\gamma }^{m-1}}& {\left[w_m\right]}^{{\gamma }^{m-2}}& & \left[w_1\right]\end{array}\end{array}\right].\end{array}$$

Applications

Finally, we use theorem 3 to compute the matrix representations of $F{G}_1$ and $F{G}_2$, when the prime $p=3$.

1) $G_1=⟨\alpha ,\beta :{\alpha }^{3^2}={\beta }^3=1,\beta \alpha ={\alpha }^{1+3}\beta ⟩$

$=\{1,\alpha ,{\alpha }^2,\dots ,{\alpha }^8,\beta ,\alpha \beta ,{\alpha }^2\beta ,\dots ,{\alpha }^8\beta ,{\beta }^2,\alpha {\beta }^2,{\alpha }^2{\beta }^2,\dots ,{\alpha }^8{\beta }^2\}$.

The general element of $F{G}_1$ is $w=a_{0}1+a_1\alpha +\dots +a_8{\alpha }^8+a_9\beta +a_{10}\alpha \beta +\dots +a_{17}{\alpha }^8\beta +a_{18}{\beta }^2+a_{19}\alpha {\beta }^2+\dots +a_{26}{\alpha }^8{\beta }^2$. Let $w_1=a_{0}1+a_1\alpha +\dots +a_8{\alpha }^8$, $w_2=a_9\beta +a_{10}\alpha \beta +\dots +a_{17}{\alpha }^8\beta$, $w_3=a_{18}{\beta }^2+a_{19}\alpha {\beta }^2+\dots +a_{26}{\alpha }^8{\beta }^2$. Then $w=w_1+w_2+w_3$.

By theorem 3, matrix representation of $w$ is $\left[w\right]=\left[\begin{array}{ccc}\left[w_1\right]& {\left[w_3\right]}^{{\beta }^2}& {\left[w_2\right]}^{\beta }\\ {\left[w_2\right]}^{\beta }& \left[w_1\right]& {\left[w_3\right]}^{{\beta }^2}\\ {\left[w_3\right]}^{{\beta }^2}& {\left[w_2\right]}^{\beta }& \left[w_1\right]\end{array}\right]$

$$[w_1]=\left[\begin{array}{lllllllll}{a}_0& a_8& a_7& a_6& a_5& a_4& a_3& a_2& a_1\\ a_1& a_0& a_8& a_7& a_6& a_5& a_4& a_3& a_2\\ a_2& a_1& a_0& a_8& a_7& a_6& a_5& a_4& a_3\\ a_3& a_2& a_1& a_0& a_8& a_7& a_6& a_5& a_4\\ a_4& a_3& a_2& a_1& a_0& a_8& a_7& a_6& a_5\\ a_5& a_4& a_3& a_2& a_1& a_0& a_8& a_7& a_6\\ a_6& a_5& a_4& a_3& a_2& a_1& a_0& a_8& a_7\\ a_7& a_6& a_5& a_4& a_3& a_2& a_1& a_0& a_8\\ a_8& a_7& a_6& a_5& a_4& a_3& a_2& a_1& a_0\end{array}\right]$$

$G_1=C_9⟨\alpha ⟩{⋊}_{\phi }{C}_3⟨\beta ⟩$, $\phi :C_3⟨\beta ⟩\to Aut(C_9⟨\alpha ⟩)$ is a homomorphism such that $\phi \left(\beta \right)\left(\alpha \right)=\beta {\alpha }^3\beta$

$\phi \left(\beta \right)\left(1\right)=\beta 1{\beta }^{-1}=1$, $\phi \left(\beta \right)\left(\alpha \right)=\beta \alpha {\beta }^{-1}={\alpha }^4$, $\phi \left(\beta \right)\left({\alpha }^2\right)=\beta {\alpha }^2{\beta }^{-1}={\alpha }^8$, $\phi \left(\beta \right)\left({\alpha }^3\right)=\beta {\alpha }^3\beta ={\alpha }^3$, $\phi \left(\beta \right)\left({\alpha }^4\right)=\beta {\alpha }^4{\beta }^{-1}={\alpha }^7$, $\phi \left(\beta \right)\left({\alpha }^5\right)=\beta {\alpha }^5{\beta }^{-1}={\alpha }^2$, $\phi \left(\beta \right)\left({\alpha }^6\right)=\beta {\alpha }^6{\beta }^{-1}={\alpha }^6$, $\phi \left(\beta \right)\left({\alpha }^7\right)=\beta {\alpha }^7{\beta }^{-1}=\alpha$, $\phi \left(\beta \right)\left({\alpha }^8\right)=\beta {\alpha }^8{\beta }^{-1}={\alpha }^5$.

$$\begin{array}{r}\left[w_2\right]=\left[col\left(1\right)\left|col\left(\alpha \right)\right|col\left({\alpha }^2\right)\left|col\left({\alpha }^3\right)\right|col\left({\alpha }^4\right)\left|col\left({\alpha }^5\right)\right|col\left({\alpha }^6\right)|col\left({\alpha }^7\right)|col\left({\alpha }^8\right)\right]\end{array}$$

$$\begin{array}{r}{\left[w_2\right]}^{\beta }=\left[col\left(1\right)\left|col\left({\alpha }^4\right)\right|col\left({\alpha }^8\right)\left|col\left({\alpha }^3\right)\right|col\left({\alpha }^7\right)\left|col\left({\alpha }^2\right)\right|col\left({\alpha }^6\right)|col\left(\alpha \right)|col\left({\alpha }^5\right)\right]\end{array}$$

$$[w_2{]}^{\beta }=\left[\begin{array}{c}\begin{array}{ccccccccc}{a}_9& a_{14}& a_{10}|& a_{15}& a_{11}& a_{16}|& a_{12}& a_{17}& a_{13}\\ a_{10}& a_{15}& a_{11}|& a_{16}& a_{12}& a_{17}|& a_{13}& a_9& a_{14}\\ a_{11}& a_{16}& a_{12}|& a_{17}& a_{13}& a_9|& a_{14}& a_{10}& a_{15}\end{array}\\ –––––––––––––––––––––––\\ \begin{array}{ccccccccc}{a}_{12}& a_{17}& a_{13}|& a_9& a_{14}& a_{10}|& a_{15}& a_{11}& a_{16}\\ a_{13}& a_9& a_{14}|& a_{10}& a_{15}& a_{11}|& a_{16}& a_{12}& a_{17}\\ a_{14}& a_{10}& a_{15}|& a_{11}& a_{16}& a_{12}|& a_{17}& a_{13}& a_9\end{array}\\ –––––––––––––––––––––––\\ \begin{array}{ccccccccc}{a}_{15}& a_{11}& a_{16}|& a_{12}& a_{17}& a_{13}|& a_9& a_{14}& a_{10}\\ a_{16}& a_{12}& a_{17}|& a_{13}& a_9& a_{14}|& a_{10}& a_{15}& a_{11}\\ a_{17}& a_{13}& a_9|& a_{14}& a_{10}& a_{15}|& a_{11}& a_{16}& a_{12}\end{array}\end{array}\right]$$

$$\begin{array}{r}\phi \left({\beta }^2\right)\left(\alpha \right)={\beta }^2\alpha {\beta }^{-2}\end{array}$$

$\phi \left({\beta }^2\right)\left(1\right)={\beta }^{2}1{\beta }^{-2}=1$, $\phi \left({\beta }^2\right)\left(\alpha \right)={\beta }^2\alpha {\beta }^{-2}={\alpha }^7$, $\phi \left({\beta }^2\right)\left({\alpha }^2\right)={\beta }^2{\alpha }^2{\beta }^{-2}={\alpha }^5$, $\phi \left({\beta }^2\right)\left({\alpha }^3\right)={\beta }^2{\alpha }^3{\beta }^{-2}={\alpha }^3$, $\phi \left({\beta }^2\right)\left({\alpha }^4\right)={\beta }^2{\alpha }^4{\beta }^{-2}=\alpha ,\phi \left({\beta }^2\right)\left({\alpha }^5\right)={\beta }^2{\alpha }^5{\beta }^{-2}={\alpha }^8$, $\phi \left({\beta }^6\right)\left({\alpha }^6\right)={\beta }^2{\alpha }^6{\beta }^{-2}={\alpha }^6$, $\phi \left({\beta }^2\right)\left({\alpha }^7\right)={\beta }^2{\alpha }^7{\beta }^{-2}={\alpha }^4$, $\phi \left({\beta }^2\right)\left({\alpha }^8\right)={\beta }^2{\alpha }^8{\beta }^{-2}={\alpha }^2$.

$$\begin{array}{r}\left[w_3\right]=\left[col\left(1\right)\left|col\left(\alpha \right)\right|col\left({\alpha }^2\right)\left|col\left({\alpha }^3\right)\right|col\left({\alpha }^4\right)\left|col\left({\alpha }^5\right)\right|col\left({\alpha }^6\right)|col\left({\alpha }^7\right)|col\left({\alpha }^8\right)\right]\end{array}$$

$$\begin{array}{r}{\left[w_3\right]}^{{\beta }^2}=\left[col\left(1\right)\left|col\left({\alpha }^7\right)\right|col\left({\alpha }^5\right)\left|col\left({\alpha }^3\right)\right|col\left(\alpha \right)\left|col\left({\alpha }^8\right)\right|col\left({\alpha }^6\right)|col\left({\alpha }^4\right)|col\left({\alpha }^2\right)\right]\end{array}$$

$$[w_3{]}^{{\beta }^2}=\left[\begin{array}{c}\begin{array}{ccccccccc}{a}_{18}& a_{20}& a_{22}|& a_{24}& a_{26}& a_{19}|& a_{21}& a_{23}& a_{25}\\ a_{19}& a_{21}& a_{23}|& a_{25}& a_{18}& a_{20}|& a_{22}& a_{24}& a_{26}\\ a_{20}& a_{22}& a_{24}|& a_{26}& a_{19}& a_{21}|& a_{23}& a_{25}& a_{18}\end{array}\\ –––––––––––––––––––––––\\ \begin{array}{ccccccccc}{a}_{21}& a_{23}& a_{25}|& a_{18}& a_{20}& a_{22}|& a_{24}& a_{26}& a_{19}\\ a_{22}& a_{24}& a_{26}|& a_{19}& a_{21}& a_{23}|& a_{25}& a_{18}& a_{20}\\ a_{23}& a_{25}& a_{18}|& a_{20}& a_{22}& a_{24}|& a_{26}& a_{19}& a_{21}\end{array}\\ –––––––––––––––––––––––\\ \begin{array}{ccccccccc}{a}_{24}& a_{26}& a_{19}|& a_{21}& a_{23}& a_{25}|& a_{18}& a_{20}& a_{22}\\ a_{25}& a_{18}& a_{20}|& a_{22}& a_{24}& a_{26}|& a_{19}& a_{21}& a_{23}\\ a_{26}& a_{19}& a_{21}|& a_{23}& a_{25}& a_{18}|& a_{20}& a_{22}& a_{24}\end{array}\end{array}\right]$$

2) $G_2=⟨\alpha ,\beta ,\gamma :{\alpha }^3={\beta }^3={\gamma }^3=1,\alpha \beta =\beta \alpha ,\gamma \beta =\beta \gamma ,\gamma \alpha =\alpha \beta \gamma ⟩$ $G_2=\{1,\alpha ,{\alpha }^2,\beta ,\alpha \beta ,{\alpha }^2\beta ,{\beta }^2,\alpha {\beta }^2,{\alpha }^2{\beta }^2,\gamma ,\alpha \gamma ,{\alpha }^2\gamma ,\beta \gamma ,\alpha \beta \gamma ,{\alpha }^2\beta \gamma ,{\beta }^2\gamma ,\alpha {\beta }^2\gamma ,$ ${\alpha }^2{\beta }^2\gamma ,{\gamma }^2,\alpha {\gamma }^2,{\alpha }^2{\gamma }^2,\beta {\gamma }^2,\alpha \beta {\gamma }^2,{\alpha }^2\beta {\gamma }^2,{\beta }^2{\gamma }^2,\alpha {\beta }^2{\gamma }^2,{\alpha }^2{\beta }^2{\gamma }^2\}$

The general element of $F{G}_2$ is $w=a_{0}1+a_1\alpha +a_2{\alpha }^2+a_3\beta +a_4\alpha \beta +a_5{\alpha }^2\beta +a_6{\beta }^2+a_7\alpha {\beta }^2+a_8{\alpha }^2{\beta }^2+a_9\gamma +a_{10}\alpha \gamma +a_{11}{\alpha }^2\gamma +a_{12}\beta \gamma +a_{13}\alpha \beta \gamma +a_{14}{\alpha }^2\beta \gamma +a_{15}{\beta }^2\gamma +a_{16}\alpha {\beta }^2\gamma +a_{17}{\alpha }^2{\beta }^2\gamma +a_{18}{\gamma }^2+a_{19}\alpha {\gamma }^2+a_{20}{\alpha }^2{\gamma }^2+a_{21}\beta {\gamma }^2+a_{22}\alpha \beta {\gamma }^2+a_{23}{\alpha }^2\beta {\gamma }^2+a_{24}{\beta }^2{\gamma }^2+a_{25}\alpha {\beta }^2{\gamma }^2+a_{26}{\alpha }^2{\beta }^2{\gamma }^2$.

$$\begin{array}{r}{w}_1=a_{0}1+a_1\alpha +a_2{\alpha }^2+a_3\beta +a_4\alpha \beta +a_5{\alpha }^2\beta +a_6{\beta }^2+a_7\alpha {\beta }^2+a_8{\alpha }^2{\beta }^2\end{array}$$

$$\begin{array}{r}{w}_2=a_9\gamma +a_{10}\alpha \gamma +a_{11}{\alpha }^2\gamma +a_{12}\beta \gamma +a_{13}\alpha \beta \gamma +a_{14}{\alpha }^2\beta \gamma +a_{15}{\beta }^2\gamma +a_{16}\alpha {\beta }^2\gamma \end{array}$$

$$\begin{array}{r}{w}_3=a_{18}{\gamma }^2+a_{19}\alpha {\gamma }^2+a_{20}{\alpha }^2{\gamma }^2+a_{21}\beta {\gamma }^2+a_{22}\alpha \beta {\gamma }^2+a_{23}{\alpha }^2\beta {\gamma }^2+a_{24}{\beta }^2{\gamma }^2+a_{25}\alpha {\beta }^2{\gamma }^2+a_{26}{\alpha }^2{\beta }^2{\gamma }^2.\end{array}$$

Then $w=w_1+w_2+w_3$.

The matrix representation of $w$ is $\left[w\right]=\left[\begin{array}{ccc}\left[w_1\right]& {\left[w_3\right]}^{{\gamma }^2}& {\left[w_2\right]}^{\gamma }\\ {\left[w_2\right]}^{\gamma }& \left[w_1\right]& {\left[w_3\right]}^{{\gamma }^2}\\ {\left[w_3\right]}^{{\gamma }^2}& {\left[w_2\right]}^{\gamma }& \left[w_1\right]\end{array}\right]$

$$[w_1]=\left[\begin{array}{c}\begin{array}{lllllllll}{a}_0& a_2& a_1|& a_6& a_8& a_7|& a_3& a_5& a_4\\ a_1& a_0& a_2|& a_7& a_6& a_8|& a_4& a_3& a_5\\ a_2& a_1& a_0|& a_8& a_7& a_6|& a_5& a_4& a_3\end{array}\\ –––––––––––––––––––––––\\ \begin{array}{ccccccccc}{a}_3& a_5& a_4|& a_0& a_2& a_1|& a_6& a_8& a_7\\ a_4& a_3& a_5|& a_1& a_0& a_2|& a_7& a_6& a_8\\ a_5& a_4& a_3|& a_2& a_1& a_0|& a_8& a_7& a_6\end{array}\\ –––––––––––––––––––––––\\ \begin{array}{ccccccccc}{a}_6& a_8& a_7|& a_3& a_5& a_4|& a_0& a_2& a_1\\ a_7& a_6& a_8|& a_4& a_3& a_5|& a_1& a_0& a_2\\ a_8& a_7& a_6|& a_5& a_4& a_3|& a_2& a_1& a_0\end{array}\end{array}\right]$$

$G_2=(C_3⟨\alpha ⟩\times C_3⟨\beta ⟩){⋊}_{\phi }{C}_3⟨\gamma ⟩$, $\phi :C_3⟨\gamma ⟩\to Aut(C_3⟨\alpha ⟩\times C_3⟨\beta ⟩)$ is a homomorphism such that $\phi \left(\gamma \right)\left(\alpha \right)=\gamma \alpha {\gamma }^{-1}$.

$\phi \left(\gamma \right)\left(1\right)=\gamma 1{\gamma }^{-1}=1$, $\phi \left(\gamma \right)\left(\alpha \right)=\gamma \alpha {\gamma }^{-1}=\alpha \beta$, $\phi \left(\gamma \right)\left({\alpha }^2\right)=\gamma {\alpha }^2{\gamma }^{-1}={\alpha }^2{\beta }^2$, $\phi \left(\gamma \right)\left(\beta \right)=\gamma \beta {\gamma }^{-1}=\beta$, $\phi \left(\gamma \right)\left(\alpha \beta \right)=\gamma \alpha \beta {\gamma }^{-1}=\alpha {\beta }^2$, $\phi \left(\gamma \right)\left({\alpha }^2\beta \right)=\gamma {\alpha }^2\beta {\gamma }^{-1}={\alpha }^2$, $\phi \left(\gamma \right)\left({\beta }^2\right)=\gamma {\beta }^2{\gamma }^{-1}={\beta }^2$, $\phi \left(\gamma \right)\left(\alpha {\beta }^2\right)=\gamma \alpha {\beta }^2{\gamma }^{-1}=\alpha$, $\phi \left(\gamma \right)\left({\alpha }^2{\beta }^2\right)=\gamma {\alpha }^2{\beta }^2{\gamma }^{-1}={\alpha }^2\beta$.

$$\begin{array}{r}\left[w_2\right]=\left[col\left(1\right)\left|col\left(\alpha \right)\right|col\left({\alpha }^2\right)\left|col\left(\beta \right)\right|col\left(\alpha \beta \right)\left|col\left({\alpha }^2\beta \right)\right|col\left({\beta }^2\right)|col\left(\alpha {\beta }^2\right)|col\left({\alpha }^2{\beta }^2\right)\right]\end{array}$$

$$\begin{array}{r}{\left[w_2\right]}^{\gamma }=\left[col\left(1\right)\left|col\left(\alpha \beta \right)\right|col\left({\alpha }^2{\beta }^2\right)\left|col\left(\beta \right)\right|col\left(\alpha {\beta }^2\right)\left|col\left({\alpha }^2\right)\right|col\left({\beta }^2\right)|col\left(\alpha \right)|col\left({\alpha }^2\beta \right)\right]\end{array}$$

$$[w_2{]}^{\gamma }=\left[\begin{array}{c}\begin{array}{lllllllll}{a}_9& a_{17}& a_{13}|& a_{15}& a_{14}& a_{10}|& a_{12}& a_{11}& a_{16}\\ a_{10}& a_{15}& a_{14}|& a_{16}& a_{12}& a_{11}|& a_{13}& a_9& a_{17}\\ a_{11}& a_{16}& a_{12}|& a_{17}& a_{13}& a_9|& a_{14}& a_{10}& a_{15}\end{array}\\ –––––––––––––––––––––––\\ \begin{array}{ccccccccc}{a}_{12}& a_{11}& a_{16}|& a_9& a_{17}& a_{13}|& a_{15}& a_{14}& a_{10}\\ a_{13}& a_9& a_{17}|& a_{10}& a_{15}& a_{14}|& a_{16}& a_{12}& a_{11}\\ a_{14}& a_{10}& a_{15}|& a_{11}& a_{16}& a_{12}|& a_{17}& a_{13}& a_9\end{array}\\ –––––––––––––––––––––––\\ \begin{array}{ccccccccc}{a}_{15}& a_{14}& a_{10}|& a_{12}& a_{11}& a_{16}|& a_9& a_{17}& a_{13}\\ a_{16}& a_{12}& a_{11}|& a_{13}& a_9& a_{17}|& a_{10}& a_{15}& a_{14}\\ a_{17}& a_{13}& a_9|& a_{14}& a_{10}& a_{15}|& a_{11}& a_{16}& a_{12}\end{array}\end{array}\right]$$

$$\begin{array}{r}\phi \left({\gamma }^2\right)\left(\alpha \right)={\gamma }^2\alpha {\gamma }^{-2}\end{array}$$

$\phi \left({\gamma }^2\right)\left(1\right)={\gamma }^{2}1{\gamma }^{-2}=1$, $\phi \left({\gamma }^2\right)\left(\alpha \right)={\gamma }^2\alpha {\gamma }^{-2}=\alpha {\beta }^2$, $\phi \left({\gamma }^2\right)\left({\alpha }^2\right)={\gamma }^2{\alpha }^2{\gamma }^{-2}={\alpha }^2\beta$, $\phi \left({\gamma }^2\right)\left(\beta \right)={\gamma }^2\beta {\gamma }^{-2}=\beta$, $\phi \left({\gamma }^2\right)\left(\alpha \beta \right)={\gamma }^2\alpha \beta {\gamma }^{-2}=\alpha$, $\phi \left({\gamma }^2\right)\left({\alpha }^2\beta \right)={\gamma }^2{\alpha }^2\beta {\gamma }^{-2}={\alpha }^2{\beta }^2$, $\phi \left({\gamma }^2\right)\left({\beta }^2\right)={\gamma }^2{\beta }^2{\gamma }^{-2}={\beta }^2$, $\phi \left({\gamma }^2\right)\left(\alpha {\beta }^2\right)={\gamma }^2\alpha {\beta }^2{\gamma }^{-2}=\alpha \beta$, $\phi \left({\gamma }^2\right)\left({\alpha }^2{\beta }^2\right)={\gamma }^2{\alpha }^2{\beta }^2{\gamma }^{-2}={\alpha }^2$.

$$\begin{array}{r}[w_3]=[col\left(1\right)\left|col\left(\alpha \right)\right|col\left({\alpha }^2\right)\left|col\left(\beta \right)\right|col\left(\alpha \beta \right)\left|col\left({\alpha }^2\beta \right)\right|col\left({\beta }^2\right)|col\left(\alpha {\beta }^2\right)|col({\alpha }^2{\beta }^2)]\end{array}$$

$$\begin{array}{r}{\left[w_3\right]}^{{\gamma }^2}=[col\left(1\right)\left|col\left(\alpha {\beta }^2\right)\right|col\left({\alpha }^2\beta \right)\left|col\left(\beta \right)\right|col\left(\alpha \right)\left|col\left({\alpha }^2{\beta }^2\right)\right|col\left({\beta }^2\right)|col\left(\alpha \beta \right)|col({\alpha }^2)]\end{array}$$

$$[w_3{]}^{{\gamma }^2}=\left[\begin{array}{c}\begin{array}{lllllllll}{a}_{18}& a_{23}& a_{25}|& a_{24}& a_{20}& a_{22}|& a_{21}& a_{26}& a_{19}\\ a_{19}& a_{21}& a_{26}|& a_{25}& a_{18}& a_{23}|& a_{22}& a_{24}& a_{20}\\ a_{20}& a_{22}& a_{24}|& a_{26}& a_{19}& a_{21}|& a_{23}& a_{25}& a_{18}\end{array}\\ –––––––––––––––––––––––\\ \begin{array}{ccccccccc}{a}_{21}& a_{26}& a_{19}|& a_{18}& a_{23}& a_{25}|& a_{24}& a_{20}& a_{22}\\ a_{22}& a_{24}& a_{20}|& a_{19}& a_{21}& a_{26}|& a_{25}& a_{18}& a_{23}\\ a_{23}& a_{25}& a_{18}|& a_{20}& a_{22}& a_{24}|& a_{26}& a_{19}& a_{21}\end{array}\\ –––––––––––––––––––––––\\ \begin{array}{ccccccccc}{a}_{24}& a_{20}& a_{22}|& a_{21}& a_{26}& a_{19}|& a_{18}& a_{23}& a_{25}\\ a_{25}& a_{18}& a_{23}|& a_{22}& a_{24}& a_{20}|& a_{19}& a_{21}& a_{26}\\ a_{26}& a_{19}& a_{21}|& a_{23}& a_{25}& a_{18}|& a_{20}& a_{22}& a_{24}\end{array}\end{array}\right]$$

For greater prime $p$, the same method may be applied.

Conflict of Interest

Conflict of Interest: Authors declare that they do not have any conflict of interest.