Subgroup ($H$) information
| Description: | $C_6^4:D_6$ |
| Order: | \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,14,4)(3,9,12)(5,17,15)(10,18,13), (20,25)(21,24), (20,24)(21,25), (1,17,6,4,15,16,14,5,7) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^6.S_4\wr C_2$ |
| Order: | \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^4.C_3^4.C_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_6^4.C_6^2.C_2^2$ |
| $W$ | $C_6^4:D_6$, of order \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4:D_6$ |
| Normal closure: | $C_3^6.\POPlus(4,3)$ |
| Core: | $C_2\times C_6^3$ |
Other information
| Number of subgroups in this autjugacy class | $54$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.S_4\wr C_2$ |