Subgroup ($H$) information
| Description: | $C_2^3\times S_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(405\)\(\medspace = 3^{4} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(2,3)(5,9)(7,8)(11,13)(12,14), (1,4)(2,7), (1,8)(2,7)(3,4)(5,6), (2,7), (1,8,2)(3,7,4)(5,9,6), (1,8,2,4,3,7)(5,9,6)(11,14)(12,13), (2,7)(3,8)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $A_5\times S_3\wr S_3$ |
| Order: | \(77760\)\(\medspace = 2^{6} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\wr S_3\times S_5$, of order \(155520\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_4\times C_2^3:\GL(3,2)$, of order \(32256\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7 \) |
| $W$ | $C_3\times S_4$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $135$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $A_5\times S_3\wr S_3$ |