Subgroup ($H$) information
| Description: | $C_2^5.C_6^2$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(8,15)(9,10)(11,13)(12,14), (1,3,2)(4,6)(5,7)(8,15)(9,10)(11,13)(12,14) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $A_4\times D_4^2:S_3$ |
| Order: | \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:(C_2^2\times C_4)$, of order \(73728\)\(\medspace = 2^{13} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2^2.D_6^2.C_2^5$ |
| $\operatorname{res}(S)$ | $C_{10}:D_5^3$, of order \(10000\)\(\medspace = 2^{4} \cdot 5^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $C_2^3\times A_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $(\SO(3,7)\times S_4^2).C_2^2$ |