Subgroup ($H$) information
| Description: | $C_{15}$ |
| Order: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Index: | \(4160\)\(\medspace = 2^{6} \cdot 5 \cdot 13 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{lll}\alpha^{5} & \alpha^{6} & \alpha^{3} \\ \alpha^{3} & \alpha & 1 \\ \alpha^{3} & \alpha^{3} & \alpha^{7} \\ \end{array}\right), \left(\begin{array}{lll}\alpha & \alpha^{4} & \alpha^{3} \\ \alpha & \alpha^{7} & \alpha^{14} \\ \alpha^{3} & \alpha^{2} & \alpha^{14} \\ \end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $\SU(3,4)$ |
| Order: | \(62400\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| Exponent: | \(780\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 13 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGammaU(3,4)$, of order \(249600\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{15}$ | |
| Normalizer: | $C_5\times S_3$ | |
| Normal closure: | $\SU(3,4)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_5\times A_4$ | $C_5\times S_3$ |
| Maximal under-subgroups: | $C_5$ | $C_3$ |
Other information
| Number of subgroups in this conjugacy class | $2080$ |
| Möbius function | $0$ |
| Projective image | $\SU(3,4)$ |