Subgroup ($H$) information
| Description: | $C_3:S_3$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{189} & \alpha^{196} \\ \alpha^{175} & \alpha^{241} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{121} & 0 \\ \alpha^{171} & 1 \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{88} & \alpha^{141} \\ \alpha^{120} & \alpha^{10} \\ \end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $C_3^4:S_3$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\AGL(5,3)$, of order \(115562653240320\)\(\medspace = 2^{10} \cdot 3^{15} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8188128\)\(\medspace = 2^{5} \cdot 3^{9} \cdot 13 \) |
| $W$ | $C_3:S_3$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_1$ | |
| Normalizer: | $C_3:S_3$ | |
| Normal closure: | $C_3^4:S_3$ | |
| Core: | $C_3^2$ | |
| Minimal over-subgroups: | $C_3^2:S_3$ | |
| Maximal under-subgroups: | $C_3^2$ | $S_3$ |
Other information
| Number of subgroups in this autjugacy class | $32670$ |
| Number of conjugacy classes in this autjugacy class | $1210$ |
| Möbius function | $-27$ |
| Projective image | $C_3^4:S_3$ |