Subgroup ($H$) information
| Description: | $A_6.C_2$ |
| Order: | \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Index: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,37)(2,41)(3,28,13,33)(4,38,20,29)(5,31,8,32)(6,27)(7,34,19,24)(9,42,17,22) \!\cdots\! \rangle$
|
| Derived length: | $1$ |
The subgroup is maximal, nonabelian, almost simple, and nonsolvable.
Ambient group ($G$) information
| Description: | $\PSL(3,4):C_2$ |
| Order: | \(40320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, nonsolvable, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(3,4):D_6$, of order \(241920\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| $W$ | $A_6.C_2$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $56$ |
| Möbius function | $-1$ |
| Projective image | $\PSL(3,4):C_2$ |