Subgroup ($H$) information
| Description: | $C_6^2:C_2^2$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Index: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,8,9)(2,3,5)(4,6,7), (1,6,5)(2,8,7)(3,9,4), (13,14)(16,17), (10,12)(16,17), (10,12)(11,15)(13,14)(16,17), (2,4)(3,7)(5,6)(8,9)\rangle$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_6^2:C_2^2.S_4^2$ |
| Order: | \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $S_4^2$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $S_4\wr C_2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^9.A_4\wr C_3$, of order \(995328\)\(\medspace = 2^{12} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $\GL(2,3).C_2^6.\GL(3,2)$, of order \(580608\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 7 \) |
| $W$ | $S_4\times C_3^2:\GL(2,3)$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_2^3$ |
| Normalizer: | $C_6^2:C_2^2.S_4^2$ |
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | not computed |
| Projective image | not computed |