Subgroup ($H$) information
| Description: | $C_3^2\times C_{15}$ |
| Order: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Index: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$\langle(1,3,2)(4,5,7), (4,5,7), (1,3,2)(4,7,5)(8,12,11,10,15)(9,13,14), (9,14,13)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 3$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $(C_3\times \GL(2,4)):S_4$ |
| Order: | \(12960\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.Q_8.(S_3\times S_4).S_5$ |
| $\operatorname{Aut}(H)$ | $C_4\times \GL(3,3)$, of order \(44928\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 13 \) |
| $W$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $(C_3\times \GL(2,4)):S_4$ |