Subgroup ($H$) information
| Description: | $C_4\times S_3^3$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,2,3), (1,2), (4,5,6,7), (1,3,2)(4,7,6,5)(8,14)(9,10)(11,12), (9,13), (9,13,10), (8,15,14)(9,13,10), (4,6)(5,7)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^3:(C_2^2\times D_4^2)$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_2^5.C_2^6.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times S_3^3:S_4$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $W$ | $C_2\times S_3^3$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $D_{12}:D_6^2$ |
| Normal closure: | $C_{12}:D_6^2$ |
| Core: | $C_{12}:S_3^2$ |
Other information
| Number of subgroups in this autjugacy class | $16$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | not computed |
| Projective image | $S_3^3:C_2^4$ |