Subgroup ($H$) information
| Description: | $C_2^4:A_4^2.C_{12}$ |
| Order: | \(27648\)\(\medspace = 2^{10} \cdot 3^{3} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,6)(2,3)(4,5)(7,8), (9,19)(10,18)(11,15)(13,14), (2,3)(7,8), (1,2,4,7,6,8) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^9.C_3\wr C_2^2$ |
| Order: | \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2.A_4^2.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $A_4\wr C_4.C_2^3$, of order \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2.A_4^2\wr C_2$ |
| Normal closure: | $C_2.A_4^2\wr C_2$ |
| Core: | $C_2^5.\PSOPlus(4,3)$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $A_4^2.\POPlus(4,3)$ |