Subgroup ($H$) information
| Description: | $C_3:C_8$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(81\)\(\medspace = 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a, a^{2}, a^{4}, d$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_3^3:F_9$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.C_4.C_2^2.A_6.C_2^2$, of order \(5598720\)\(\medspace = 2^{9} \cdot 3^{7} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_4$ | |
| Normalizer: | $C_3:C_8$ | |
| Normal closure: | $C_3^3:F_9$ | |
| Core: | $C_3$ | |
| Minimal over-subgroups: | $C_3:F_9$ | |
| Maximal under-subgroups: | $C_{12}$ | $C_8$ |
Other information
| Number of subgroups in this autjugacy class | $81$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $9$ |
| Projective image | $C_3^3:F_9$ |