Subgroup ($H$) information
| Description: | $C_5^2:D_{30}$ |
| Order: | \(1500\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{3} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$ad^{21}e^{14}f^{5}, e^{3}, f^{3}, d^{20}e^{5}f^{5}, cde^{5}f^{5}, d^{6}e^{3}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{15}^3.S_4$ |
| Order: | \(81000\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 5^{3} \) |
| 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 \(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $C_4\times C_5.(C_4\times S_3\times F_5)$ |
| $W$ | $D_5\times D_{15}$, of order \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $18$ |
| Möbius function | $0$ |
| Projective image | $C_{15}^3.S_4$ |