Subgroup ($H$) information
| Description: | $(A_5\times \GL(2,4)):D_4$ |
| Order: | \(86400\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5^{2} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(11,12,13), (1,8,4,10,9,6)(2,7,3,5)(11,12), (1,5)(11,13)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $S_4\times S_5\wr C_2$ |
| Order: | \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times S_5\wr C_2$, of order \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $S_5^2:D_6$, of order \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
| $W$ | $S_5^2:D_6$, of order \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $S_5^2:D_6$ |
| Normal closure: | $A_5^2.\GL(2,\mathbb{Z}/4)$ |
| Core: | $\POPlus(4,5)$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $S_4\times S_5\wr C_2$ |