Subgroup ($H$) information
| Description: | $C_3^2\times C_{15}$ |
| Order: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Index: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$\langle(1,2,3), (1,2,3)(5,9,6,8,7)(10,13,12)(11,14,15), (10,12,13)(11,14,15), (5,7,8,6,9)\rangle$
|
| 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: | $C_5\times A_4\times A_6$ |
| Order: | \(21600\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times A_6.C_2^2\times S_4$ |
| $\operatorname{Aut}(H)$ | $C_4\times \GL(3,3)$, 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 conjugacy class | $40$ |
| Möbius function | $0$ |
| Projective image | $A_4\times A_6$ |