Subgroup ($H$) information
| Description: | $C_3^2:C_8$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a^{3}, c^{2}, a^{6}, a^{12}, b$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_3\times A_4):C_{24}$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_6^2:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $C_6^2:\GL(2,3)$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_6^2:D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_3^2:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_4$ | ||
| Normalizer: | $C_3^2:C_{24}$ | ||
| Normal closure: | $C_{12}.S_4$ | ||
| Core: | $C_{12}$ | ||
| Minimal over-subgroups: | $C_{12}.S_4$ | $C_3^2:C_{24}$ | |
| Maximal under-subgroups: | $C_3\times C_{12}$ | $C_3:C_8$ | $C_3:C_8$ |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $1$ |
| Projective image | $C_3^2:S_4$ |