Subgroup ($H$) information
| Description: | $C_2^2:D_4^2\times S_3$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,2), (4,6)(5,7), (1,3,2)(4,6)(5,7)(8,15)(9,10)(11,13)(12,14), (1,2,3) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_2^6.D_6^2$ |
| Order: | \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:D_4:C_2$, of order \(221184\)\(\medspace = 2^{13} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | Group of order \(18874368\)\(\medspace = 2^{21} \cdot 3^{2} \) |
| $W$ | $C_{12}:C_2^5$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3:(C_2^5.C_2^5)$ |
| Normal closure: | $C_2^5.D_6^2$ |
| Core: | $C_2^6:D_6$ |
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | not computed |
| Projective image | $C_2^5.D_6^2$ |