Subgroup ($H$) information
| Description: | $D_5\times C_5^2$ |
| Order: | \(250\)\(\medspace = 2 \cdot 5^{3} \) |
| Index: | \(81\)\(\medspace = 3^{4} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$a^{15}, d^{3}, c^{3}d^{12}, a^{6}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Hall subgroup, supersolvable (hence solvable and 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.C_6^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $F_5\times \GL(2,5)$, of order \(9600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \) |
| $W$ | $D_5$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $(C_3\times C_{15}^2):S_3$ |