Subgroup ($H$) information
| Description: | $C_2^2\times \PSL(2,11)$ |
| Order: | \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| Index: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\langle(12,15)(13,14), (1,10,3)(2,9,6)(5,11,8)(12,13)(14,15), (2,10)(4,5)(6,7)(9,11)(12,14)(13,15), (12,13)(14,15)\rangle$
|
| Derived length: | $1$ |
The subgroup is nonabelian, an A-group, and nonsolvable.
Ambient group ($G$) information
| Description: | $S_4\times M_{11}$ |
| Order: | \(190080\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times M_{11}$, of order \(190080\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $S_3\times \PGL(2,11)$, of order \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $W$ | $S_3\times \PSL(2,11)$, of order \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $12$ |
| Möbius function | $-3$ |
| Projective image | $S_4\times M_{11}$ |