Subgroup ($H$) information
| Description: | $C_{19}$ |
| Order: | \(19\) |
| Index: | \(1825988935680\)\(\medspace = 2^{18} \cdot 3^{7} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| Exponent: | \(19\) |
| Generators: |
$\left(\begin{array}{llll}\alpha^{14} & \alpha^{49} & \alpha^{23} & \alpha^{58} \\ \alpha^{2} & \alpha^{54} & \alpha^{44} & \alpha^{16} \\ 0 & \alpha^{49} & 0 & \alpha^{58} \\ \alpha^{5} & \alpha^{57} & \alpha^{14} & \alpha^{26} \\ \end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $19$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $\SU(4,8)$ |
| Order: | \(34693789777920\)\(\medspace = 2^{18} \cdot 3^{7} \cdot 5 \cdot 7^{2} \cdot 13 \cdot 19 \) |
| Exponent: | \(933660\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 13 \cdot 19 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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 \(208162738667520\)\(\medspace = 2^{19} \cdot 3^{8} \cdot 5 \cdot 7^{2} \cdot 13 \cdot 19 \) |
| $\operatorname{Aut}(H)$ | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | $\SU(4,8)$ |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $22543073280$ |
| Möbius function | not computed |
| Projective image | not computed |