Subgroup ($H$) information
| Description: | $C_2^2\wr S_3$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,2,8,4)(3,7,5,6)(9,11)(10,12), (1,2,8,4)(3,7,5,6)(9,11), (1,8)(6,7), (2,3)(4,5)(9,11)(10,12), (1,8,7)(2,4,3), (2,5)(3,4), (2,4)(3,5), (1,7)(6,8)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_2^6:\SOPlus(4,2)$ |
| Order: | \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2.C_2^5.C_2^2$ |
| $\operatorname{Aut}(H)$ | $A_4^3.C_2^3$, of order \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_2^5.D_6^2$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $\POPlus(4,3)$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $S_4^2:C_2^2$ |