Subgroup ($H$) information
| Description: | $C_3^4:(C_6\times D_4)$ |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(7,8,9)(10,12,11), (7,12)(8,11)(9,10)(13,15), (1,4)(2,6)(7,8)(10,13,12,14,11,15) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^4:(D_4\times \GL(2,3))$ |
| Order: | \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:S_3.C_6^2.C_6.D_4.C_2$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_{12}.C_2^4$, of order \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
| $W$ | $C_3^4:(D_4\times D_6)$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3^4:(D_4\times \GL(2,3))$ |