Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(2\) |
| Generators: |
$\left(\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 0 & 0 & 2
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), the Frattini subgroup, the Fitting subgroup, the radical, the socle, cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), stem, a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $\Sp(4,3)$ |
| Order: | \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and quasisimple (hence nonsolvable and perfect).
Quotient group ($Q$) structure
| Description: | $C(2,3)$ |
| Order: | \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Automorphism Group: | $\SO(5,3)$, of order \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $0$ |
The quotient is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SO(5,3)$, of order \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $\Sp(4,3)$ | |||||
| Normalizer: | $\Sp(4,3)$ | |||||
| Minimal over-subgroups: | $C_{10}$ | $C_6$ | $C_6$ | $C_6$ | $C_2^2$ | $C_4$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $-25920$ |
| Projective image | $C(2,3)$ |