Subgroup ($H$) information
| Description: | $C_2\times C_3^2.Q_8^2$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(19,20), (2,10,15,14)(4,18,7,17)(5,6,16,8)(9,11,13,12), (3,18,17)(4,11,9) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is metacyclic, monomial, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^4:(Q_8^2.C_2^3)$ |
| Order: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_4^2.C_2^6.C_2^2$, of order \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | Group of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| $W$ | $C_6^2:\SD_{16}$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^4:(Q_8^2:C_2^2)$ |