Subgroup ($H$) information
| Description: | not computed |
| Order: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| Index: | \(2\) |
| Exponent: | not computed |
| Generators: |
$\langle(5,7,6)(10,16,14), (9,16)(10,14)(11,12)(13,15), (1,6)(2,3)(4,8)(5,7)(9,10) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $A_4^2:\POPlus(4,3)$ |
| Order: | \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
| 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.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2\wr C_2.C_4.D_4$, of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |