Subgroup ($H$) information
| Description: | $C_2\times \SD_{16}$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(2,6,7,3)(4,5,8,9), (2,5,7,9)(3,8,6,4), (2,7)(3,6)(4,8)(5,9), (2,4)(5,9)(7,8)(11,13), (10,12)(11,13)\rangle$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $D_4^2:C_2$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{res}(S)$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $27$ |
| Möbius function | $0$ |
| Projective image | $C_6^2:\GL(2,3)$ |