Subgroup ($H$) information
| Description: | $C_3\times S_4$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(10,13)(11,12), (2,4)(5,9)(7,8)(11,13), (10,12)(11,13), (1,6,3)(2,8,5)(4,7,9), (1,5,4)(2,7,6)(3,8,9)(11,13,12)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_6^2:\GL(2,3)$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:S_3.C_2^2:S_4:C_2$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\operatorname{res}(S)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3$ | |||
| Normalizer: | $C_3\times S_4$ | |||
| Normal closure: | $C_6^2:\GL(2,3)$ | |||
| Core: | $C_2^2$ | |||
| Minimal over-subgroups: | $C_3^2:S_4$ | |||
| Maximal under-subgroups: | $C_3\times A_4$ | $C_3\times D_4$ | $S_4$ | $C_3\times S_3$ |
Other information
| Number of subgroups in this conjugacy class | $24$ |
| Möbius function | $0$ |
| Projective image | $C_6^2:\GL(2,3)$ |