Subgroup ($H$) information
| Description: | $C_3\times C_{15}$ |
| Order: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| Index: | \(450\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$a^{20}, a^{6}c^{6}d^{6}, d^{10}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.C_6^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_4\times \GL(2,3)$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $72$ |
| Number of conjugacy classes in this autjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $C_{15}\wr S_3$ |