Subgroup ($H$) information
| Description: | $C_2^2\times F_5\times S_5$ |
| Order: | \(9600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,4)(3,5), (1,11,13,12,14)(2,4)(6,7)(8,10,9), (1,12)(2,4)(11,13), (1,13,12,11), (2,4), (1,12,14,11)(2,4)(6,10), (1,12,11,14,13)(2,4)(3,5)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $S_5^2\times D_4$ |
| Order: | \(115200\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, nonsolvable, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^2\times D_4).A_5^2.D_4$ |
| $\operatorname{Aut}(H)$ | $F_5.S_5\times C_2^2:S_4$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | not computed |
| Projective image | not computed |