Subgroup ($H$) information
Description: | $C(2,3)$ |
Order: | \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \) |
Index: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
Generators: | $\left[ \left(\begin{array}{rrrr} 2 z_{2} + 2 & 0 & 2 z_{2} + 1 & 2 z_{2} \\ 2 z_{2} + 1 & z_{2} + 1 & 1 & 2 z_{2} + 2 \\ 2 & 0 & z_{2} & 0 \\ 2 z_{2} + 1 & 0 & 2 z_{2} + 2 & z_{2} + 1 \end{array}\right) \right], \left[ \left(\begin{array}{rrrr} 2 z_{2} + 2 & z_{2} + 1 & 2 & 2 z_{2} \\ 2 z_{2} & 2 z_{2} + 1 & z_{2} + 1 & 2 z_{2} \\ 2 z_{2} + 2 & 2 z_{2} + 2 & 2 z_{2} + 1 & 0 \\ 2 z_{2} + 2 & z_{2} + 1 & z_{2} + 1 & 2 \end{array}\right) \right]$ |
Derived length: | $0$ |
The subgroup is maximal, nonabelian, and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Ambient group ($G$) information
Description: | ${}^2A(3,3)$ |
Order: | \(3265920\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 5 \cdot 7 \) |
Exponent: | \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
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 $W = N_G(H) / Z_G(H)$.
$\operatorname{Aut}(G)$ | $\PGammaU(4,3)$, of order \(26127360\)\(\medspace = 2^{10} \cdot 3^{6} \cdot 5 \cdot 7 \) |
$\operatorname{Aut}(H)$ | $\SO(5,3)$, of order \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
$W$ | $C(2,3)$, of order \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \) |
Related subgroups
Other information
Number of subgroups in this conjugacy class | $126$ |
Möbius function | not computed |
Projective image | ${}^2A(3,3)$ |