Subgroup ($H$) information
| Description: | $A_4^2:\POPlus(4,3)$ |
| Order: | \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(5,9,15)(6,14,8)(10,12,16), (10,12,16), (3,4)(11,13), (6,14)(9,15)(10,12) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is normal, nonabelian, solvable, and rational. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $A_4^2.S_4\wr C_2.C_4$ |
| Order: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| 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: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3\times C_2^8.C_3^4.D_4.C_2^2$ |
| $\operatorname{Aut}(H)$ | $A_4^2\wr C_2.C_4.D_4$, of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| $W$ | $A_4^2:\POPlus(4,3).C_2^2$, of order \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $A_4^2.S_4\wr C_2.C_4$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $A_4^2.S_4\wr C_2.C_4$ |