Subgroup ($H$) information
| Description: | $C_2^7:C_6$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,7)(2,4), (1,3,4,6)(2,8,7,5)(9,15,10,16)(11,14,13,12), (3,5)(6,8), (2,4,7) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_2^2\wr S_3:C_{12}$ |
| Order: | \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^4:C_3.C_2^4.C_2^4$ |
| $\operatorname{res}(S)$ | $D_4\times C_2^4:S_4$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_4\times C_2^3:A_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |