Subgroup ($H$) information
| Description: | $C_3\times S_3$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Index: | \(243\)\(\medspace = 3^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$a^{3}, bc^{3}, c^{6}d$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_9^2.(S_3\times C_3^2)$ |
| Order: | \(4374\)\(\medspace = 2 \cdot 3^{7} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^3.C_3^3.C_6.C_2$, of order \(236196\)\(\medspace = 2^{2} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\operatorname{res}(S)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3$ | ||
| Normalizer: | $C_3\times S_3$ | ||
| Normal closure: | $(C_3\times C_9^2):C_6$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_3^2:C_6$ | $C_3^2:C_6$ | |
| Maximal under-subgroups: | $C_3^2$ | $C_6$ | $S_3$ |
Other information
| Number of subgroups in this autjugacy class | $729$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $C_9^2.(S_3\times C_3^2)$ |