Subgroup ($H$) information
| Description: | $(C_6\times \GL(2,4)):C_4$ |
| Order: | \(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(7,9), (3,5,8), (3,5,8)(7,9)(10,11)(12,13), (4,6)(5,8), (1,4,6), (1,4)(5,8)(7,9)(10,14,13), (1,8,6,5)(3,4)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, an A-group, and nonsolvable.
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 \(155520\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_9.(C_2\times S_5)$ |
| $W$ | $A_5\times \SOPlus(4,2)$, of order \(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $A_5\times S_3\wr S_3$ |