Subgroup ($H$) information
| Description: | $C_3^3:A_4$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(1,4)(2,3), (10,17,13)(12,14,16), (11,15,18)(12,14,16), (1,4,2)(5,6,7)(10,16,15,13,12,11,17,14,18), (5,6,7)(11,15,18)(12,16,14), (1,2)(3,4)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_6^3:S_3^2$ |
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6^2.C_3^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_3\times C_6^2:S_3^2$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $W$ | $C_3^2:S_4$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_6^3:S_3^2$ |