Subgroup ($H$) information
| Description: | $C_5\times C_3^3:A_4$ |
| Order: | \(1620\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5 \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Generators: |
$\langle(1,5,7)(2,9,3)(4,6,8), (1,3,6)(10,12,13,11,14), (2,8,5), (4,9,7), (1,6,3)(2,8,5), (3,6)(5,8), (4,9)(5,8)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^3:(S_4\times S_5)$ |
| 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 \(155520\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_4\times S_3\wr S_3$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| $W$ | $C_3^3:(C_4\times S_4)$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^3:(S_4\times S_5)$ |