Subgroup ($H$) information
| Description: | $C_{15}:S_3$ |
| Order: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Index: | \(11\) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$a^{5}, b, c^{22}, a^{2}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_3\times C_{33}):C_{10}$ |
| Order: | \(990\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{11}\times C_3^2:\GL(2,3)$, of order \(47520\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_4\times C_3^2:\GL(2,3)$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(10\)\(\medspace = 2 \cdot 5 \) |
| $W$ | $C_3:S_3$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $11$ |
| Möbius function | $-1$ |
| Projective image | $(C_3\times C_{33}):C_{10}$ |