Subgroup ($H$) information
| Description: | $C_5:D_{30}$ |
| Order: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Index: | \(3\) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$a, b^{10}, b^{3}, c^{15}, c^{6}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{15}:D_{30}$ |
| Order: | \(900\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \AGL(2,3)\times C_5^2:C_4.S_5$ |
| $\operatorname{Aut}(H)$ | $C_2\times S_3\times C_5^2:C_4.S_5$ |
| $\card{\operatorname{res}(S)}$ | \(144000\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_5:D_{15}$, of order \(150\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $-1$ |
| Projective image | $C_{15}:D_{15}$ |