Subgroup ($H$) information
| Description: | $C_2^2\times \PSL(2,11)$ |
| Order: | \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| Index: | \(326700\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 11^{2} \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\langle(25,28,32)(26,30,33)(27,29,31), (1,5)(2,10)(6,9)(7,8), (13,16)(14,19)(15,22)(20,21), (23,31)(24,30)(26,28)(29,33)\rangle$
|
| Derived length: | $1$ |
The subgroup is nonabelian, an A-group, and nonsolvable.
Ambient group ($G$) information
| Description: | $\PSL(2,11)\wr C_3$ |
| Order: | \(862488000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \cdot 11^{3} \) |
| Exponent: | \(990\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $1$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(3449952000\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{3} \cdot 11^{3} \) |
| $\operatorname{Aut}(H)$ | $S_3\times \PGL(2,11)$, of order \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $9075$ |
| Möbius function | not computed |
| Projective image | not computed |