Subgroup ($H$) information
| Description: | not computed |
| Order: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | not computed |
| Generators: |
$\langle(15,16)(17,18), (1,2)(5,6)(7,8)(11,12), (7,8)(11,12)(13,15,18)(14,16,17) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, metabelian (hence solvable), and an A-group. Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $A_4^2\wr C_2.C_2^2.C_2^3$ |
| Order: | \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $D_4:C_2^2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $S_4\wr C_2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Outer Automorphisms: | $\SOPlus(4,2)$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_3^4.C_2.C_2^6.C_2^4$ |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $A_4^2:\POPlus(4,3).C_2^3$, of order \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $A_4^2\wr C_2.C_2^2.C_2^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4^2:\POPlus(4,3).C_2^3$ |