Subgroup ($H$) information
| Description: | $C_3^4:(C_4\times Q_8)$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,5,8,4)(2,9,7,6)(10,17,18,14)(12,13,16,15), (11,17,18,12)(13,15,16,14) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^4:(Q_8^2:C_4)$ |
| Order: | \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SOPlus(4,2)^2.D_4$, of order \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_3:S_3.C_6^2.C_{12}.C_2^3$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $\card{W}$ | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_3^4:(Q_8^2:C_4)$ |