Subgroup ($H$) information
| Description: | $C_{19}^2:D_9$ |
| Order: | \(6498\)\(\medspace = 2 \cdot 3^{2} \cdot 19^{2} \) |
| Index: | \(14400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(342\)\(\medspace = 2 \cdot 3^{2} \cdot 19 \) |
| Generators: |
$\langle(1,7,19,16,13,5,11,12,15,8,10,18,9,3,14,2,4,17,20)(21,25,22,32,26,27,29,36,35,38,23,28,31,30,37,39,40,34,24) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $\PSL(2,19)^2.D_4$ |
| Order: | \(93571200\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 19^{2} \) |
| Exponent: | \(6840\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \) |
| Derived length: | $2$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(2,19)^2.D_4$, of order \(93571200\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 19^{2} \) |
| $\operatorname{Aut}(H)$ | $C_{19}^2:(C_9\times D_{18})$, of order \(116964\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
| $\card{W}$ | 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 | $800$ |
| Möbius function | not computed |
| Projective image | not computed |