Subgroup ($H$) information
| Description: | $(C_2^3\times C_6^2):D_{12}$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Index: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(4,11,10)(5,9,8)(13,18,19)(16,20)(17,21), (2,5)(3,11)(4,10)(8,9)(14,17,21) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $(C_2^3\times C_6^3).C_6^2:D_6$ |
| Order: | \(746496\)\(\medspace = 2^{10} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.(C_3^2\times S_3^3):D_6$, of order \(1492992\)\(\medspace = 2^{11} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_6^2.(C_2\times A_4).C_2^6.C_2$ |
| $W$ | $(C_2^3\times C_6^2):D_{12}$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $(C_2^4\times C_6^2):D_{12}$ |
| Normal closure: | $(C_2^3\times C_6^3).C_6^2:D_6$ |
| Core: | $C_1$ |
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 | $(C_2^3\times C_6^3).C_6^2:D_6$ |