Subgroup ($H$) information
| Description: | $C_5^2\times C_{15}$ |
| Order: | \(375\)\(\medspace = 3 \cdot 5^{3} \) |
| Index: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$d^{20}f^{10}, e^{3}f^{9}, f^{3}, d^{6}e^{12}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 5$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_{15}^3.(C_4\times S_4)$ |
| Order: | \(324000\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{15}\wr S_3.C_4$, of order \(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_4.\PSL(3,5)$ |
| $W$ | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_{15}^3.(C_4\times S_4)$ |