Subgroup ($H$) information
| Description: | $C_2^3:S_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,8)(6,7), (2,5)(3,4), (1,3,6,2)(4,7,5,8)(9,11)(10,12), (2,4)(3,5), (1,3,7,5,6,4)(2,8)(9,11), (1,7)(6,8), (1,7,8)(2,4,3)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_2^6:\SOPlus(4,2)$ |
| Order: | \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2.C_2^5.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2^4:S_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $48$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | not computed |
| Projective image | $C_2^6:\SOPlus(4,2)$ |