Subgroup ($H$) information
| Description: | $C_9:C_{30}$ |
| Order: | \(270\)\(\medspace = 2 \cdot 3^{3} \cdot 5 \) |
| Index: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Generators: |
$\langle(10,12,13,14,11), (1,7,6,3,2,5)(8,9), (1,3,8,2,5,9,6,7,4), (1,2,6)(3,5,7)(4,8,9), (1,6,2)(3,5,7)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
Ambient group ($G$) information
| Description: | $A_5\times {}^2G(2,3)$ |
| Order: | \(90720\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| Exponent: | \(630\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5\times {}^2G(2,3)$, of order \(181440\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_{36}:C_6$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $W$ | $C_{18}:C_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $168$ |
| Möbius function | $0$ |
| Projective image | $A_5\times {}^2G(2,3)$ |