Subgroup ($H$) information
| Description: | $C_3^3:C_6$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
2 & 0 & 1 & 0 \\
2 & 2 & 2 & 0 \\
2 & 2 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
2 & 1 & 2 & 0 \\
2 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 \\
2 & 2 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
2 & 0 & 1 & 0 \\
2 & 2 & 2 & 0 \\
0 & 0 & 0 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 2 & 2 & 1 \\
1 & 1 & 1 & 2 \\
1 & 2 & 1 & 0 \\
2 & 2 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
1 & 0 & 2 & 0 \\
1 & 1 & 1 & 0 \\
0 & 0 & 0 & 2
\end{array}\right)$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $S_3^3:C_6$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3^3:D_6$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $\operatorname{res}(S)$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_3^2:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $1$ |
| Projective image | $S_3\wr C_3$ |