Subgroup ($H$) information
| Description: | $(C_2^2\times A_4^2):C_4$ |
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(2,4,3)(5,8,6), (1,2,5,7)(3,8)(4,6)(9,11,10,16)(12,15,14,13), (2,7)(3,4) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $A_4^2:C_2^4:C_4$ |
| Order: | \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2.C_2^5.C_2^5$ |
| $\operatorname{Aut}(H)$ | $A_4^2:D_4^2$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \) |
| $W$ | $C_2\times A_4^2:C_4$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_2^3:(A_4^2:C_4)$ |