Subgroup ($H$) information
| Description: | $C_3^3:C_8$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(81\)\(\medspace = 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a, g, eh^{2}, a^{2}, a^{4}, degh^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_3^2\times \He_3).F_9$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.C_2.D_4^2$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_9:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $W$ | $C_3^3:C_4$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $C_3^3:C_8$ | ||
| Normal closure: | $(C_3^2\times \He_3).F_9$ | ||
| Core: | $C_3^3$ | ||
| Minimal over-subgroups: | $C_3^3.F_9$ | ||
| Maximal under-subgroups: | $C_3^2:C_{12}$ | $C_3^2:C_8$ | $C_3:C_8$ |
Other information
| Number of subgroups in this autjugacy class | $81$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $(C_3^2\times \He_3).F_9$ |