Subgroup ($H$) information
| Description: | $C_{15}^2:D_6$ |
| Order: | \(2700\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5^{2} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$acd^{4}, d^{3}e^{6}, b^{2}d^{5}e^{10}, d^{5}, b^{3}, e^{3}, d^{10}e^{10}$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_{15}^2:(S_3\times D_{10})$ |
| Order: | \(27000\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5^2\times C_{15}).C_6^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{15}^2.C_{12}.C_2^3$ |
| $W$ | $S_3\times C_5^2:D_6$, of order \(1800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | $1$ |
| Projective image | $C_{15}^2:(S_3\times D_{10})$ |