Subgroup ($H$) information
| Description: | $C_2^2:C_{24}$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$b^{3}, b^{12}, e, d^{3}, b^{8}, b^{6}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{12}.(S_3\times S_4)$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_6\times S_3\times A_4).C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{res}(S)$ | $C_2^5$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $9$ |
| Möbius function | $-1$ |
| Projective image | $C_3^2:\GL(2,\mathbb{Z}/4)$ |