Subgroup ($H$) information
| Description: | $C_3^3:S_3$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
35 & 31 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
25 & 24 \\
12 & 13
\end{array}\right), \left(\begin{array}{rr}
17 & 13 \\
29 & 18
\end{array}\right), \left(\begin{array}{rr}
1 & 24 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
12 & 1
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is nonabelian and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^4:S_4$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^3.S_3^2$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| $\operatorname{res}(S)$ | $C_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_3^3:S_3$, of order \(162\)\(\medspace = 2 \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $1$ |
| Projective image | $C_3^4:S_4$ |