Subgroup ($H$) information
| Description: | $S_4\times S_5\wr C_2$ |
| Order: | \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \) |
| Index: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,7)(8,15), (3,12)(10,14), (2,14,7,12,8,6,9,10,15,3)(4,11)(5,13)(17,21) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, nonsolvable, and rational.
Ambient group ($G$) information
| Description: | $S_5^3.C_3^2:S_3$ |
| Order: | \(93312000\)\(\medspace = 2^{10} \cdot 3^{6} \cdot 5^{3} \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5^3.C_3^2:D_6$, of order \(186624000\)\(\medspace = 2^{11} \cdot 3^{6} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $S_4\times S_5\wr C_2$, of order \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \) |
| $\card{W}$ | 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 | $45$ |
| Möbius function | not computed |
| Projective image | not computed |