Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(44730\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
26 & 22 \\
37 & 45
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
63 & 10 \\
62 & 20
\end{array}\right) \right]$
|
| 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) and a $p$-group.
Ambient group ($G$) information
| Description: | $\PSL(2,71)$ |
| Order: | \(178920\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \) |
| Exponent: | \(89460\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGL(2,71)$, of order \(357840\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{36}$ | |
| Normalizer: | $D_{36}$ | |
| Normal closure: | $\PSL(2,71)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_{12}$ | $D_4$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $2485$ |
| Möbius function | $0$ |
| Projective image | $\PSL(2,71)$ |