Subgroup ($H$) information
| Description: | $C_4^2$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(3900\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{lll}\alpha & \alpha^{3} & \alpha^{13} \\ 1 & \alpha^{6} & 1 \\ \alpha^{12} & 0 & \alpha^{12} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{3} & \alpha^{4} & \alpha^{10} \\ \alpha^{5} & \alpha^{5} & \alpha \\ 1 & \alpha^{5} & \alpha^{12} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{8} & \alpha^{8} & \alpha^{12} \\ \alpha^{2} & \alpha^{11} & \alpha^{5} \\ \alpha & \alpha^{4} & \alpha^{9} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{7} & \alpha^{14} & \alpha^{5} \\ 1 & \alpha^{10} & \alpha^{11} \\ \alpha^{10} & 1 & \alpha^{13} \\ \end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 \(249600\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $\GL(2,\mathbb{Z}/4)$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_4^2$ | |
| Normalizer: | $C_4^2.A_4$ | |
| Normal closure: | $\SU(3,4)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_4^2:C_3$ | $C_4.Q_8$ |
| Maximal under-subgroups: | $C_2\times C_4$ |
Other information
| Number of subgroups in this conjugacy class | $325$ |
| Möbius function | $0$ |
| Projective image | $\SU(3,4)$ |