Subgroup ($H$) information
| Description: | $C_5^2\times D_{15}$ |
| Order: | \(750\)\(\medspace = 2 \cdot 3 \cdot 5^{3} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,7,9,10,11)(3,8,12,15,4), (2,4,9,3,11,8,7,12,10,15)(17,18), (16,18,17), (1,14,6,5,13)(3,8,12,15,4), (3,15,8,4,12)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_5^2\times C_{15}):S_4$ |
| Order: | \(9000\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3:\He_3.C_2^3$, of order \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $S_3\times \GL(2,5)\times F_5$ |
| $W$ | $S_3\times D_5$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $2$ |
| Projective image | $(C_5^2\times C_{15}):S_4$ |