Subgroup ($H$) information
| Description: | $(C_5\times A_4):S_5$ |
| Order: | \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,3,4,2)(5,7)(6,9)(10,14)(11,13)(12,15), (1,2)(3,4), (1,4)(5,8)(6,7)(10,11,15,13,12,14), (5,9,8,6,7), (1,3)(2,4), (1,2,4)\rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $(C_5\times A_4):S_6$ |
| Order: | \(43200\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(F_5\times S_4).A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $F_5\times S_4\times S_5$, of order \(57600\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \) |
| $W$ | $(C_5\times A_4):S_5$, of order \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $-1$ |
| Projective image | $(C_5\times A_4):S_6$ |