Subgroup ($H$) information
| Description: | $C_5$ |
| Order: | \(5\) |
| Index: | \(2426112\)\(\medspace = 2^{8} \cdot 3^{6} \cdot 13 \) |
| Exponent: | \(5\) |
| Generators: |
$\left[ \left(\begin{array}{rrrr}
2 & 0 & 0 & 1 \\
1 & 2 & 2 & 0 \\
0 & 2 & 2 & 2 \\
1 & 1 & 0 & 2
\end{array}\right) \right]$
|
| 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 $5$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $\PGL(4,3)$ |
| Order: | \(12130560\)\(\medspace = 2^{8} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| Exponent: | \(4680\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(4,3).C_2^2$, of order \(24261120\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{W}$ | \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_{40}$ | |||
| Normalizer: | $C_{40}:C_4$ | |||
| Normal closure: | $\PSL(4,3)$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $C_2^4:C_5$ | $C_{10}$ | $D_5$ | $D_5$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $75816$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |