Subgroup ($H$) information
| Description: | $C_{36}$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Index: | \(5402\)\(\medspace = 2 \cdot 37 \cdot 73 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
10 & 37 \\
6 & 15
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
63 & 53 \\
52 & 9
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
27 & 27 \\
32 & 5
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
60 & 61 \\
2 & 13
\end{array}\right) \right]$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $\PSL(2,73)$ |
| Order: | \(194472\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 37 \cdot 73 \) |
| Exponent: | \(97236\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \cdot 73 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGL(2,73)$, of order \(388944\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 37 \cdot 73 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{36}$ | |
| Normalizer: | $D_{36}$ | |
| Normal closure: | $\PSL(2,73)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_{73}:C_{36}$ | $D_{36}$ |
| Maximal under-subgroups: | $C_{18}$ | $C_{12}$ |
Other information
| Number of subgroups in this conjugacy class | $2701$ |
| Möbius function | $2$ |
| Projective image | $\PSL(2,73)$ |