Subgroup ($H$) information
| Description: | $A_4^3.C_2^3$ |
| Order: | \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(3,11,7)(5,12,9)(17,20)(18,19), (3,11,7), (4,9)(5,12), (1,11)(3,7), (2,6) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is normal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^9.S_3\wr S_3$ |
| Order: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2\times S_4$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^3.C_2^6:S_4$, of order \(2654208\)\(\medspace = 2^{15} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $S_4^3.S_4$, of order \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \) |
| $W$ | $A_4^3.(C_2\times S_4)$, of order \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^9.S_3\wr S_3$ |
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 | $S_4^3.S_4$ |