Subgroup ($H$) information
| Description: | $C_3\times C_{15}^2$ |
| Order: | \(675\)\(\medspace = 3^{3} \cdot 5^{2} \) |
| Index: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$a^{20}, d^{10}, a^{6}d^{3}, c^{10}, c^{3}d^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group).
Ambient group ($G$) information
| Description: | $C_{15}\wr S_3$ |
| Order: | \(20250\)\(\medspace = 2 \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.C_6^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times \SL(3,3)\times \GL(2,5)$ |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{15}^2:S_3$ |