Subgroup ($H$) information
| Description: | not computed |
| Order: | \(2048\)\(\medspace = 2^{11} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,6)(4,5)(9,16)(10,14)(11,17)(12,18), (1,5)(4,6), (5,6)(7,8), (1,7,5,3) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $A_4^2:C_2^4.S_4$ |
| Order: | \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2.A_4^2.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^6.C_2^5$ |
| Normal closure: | $A_4^2:C_2^4.S_4$ |
| Core: | $C_2^9$ |
Other information
| Number of subgroups in this autjugacy class | $27$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4^2.C_2^3:S_4$ |