Subgroup ($H$) information
| Description: | $\PSL(2,73)$ |
| Order: | \(194472\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 37 \cdot 73 \) |
| Index: | \(2\) |
| Exponent: | \(97236\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \cdot 73 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
22 & 0 \\
71 & 39
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
44 & 30 \\
6 & 68
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
27 & 0 \\
1 & 18
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
39 & 0 \\
46 & 26
\end{array}\right) \right]$
|
| Derived length: | $0$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the socle, maximal, a semidirect factor, nonabelian, and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Ambient group ($G$) information
| Description: | $\PGL(2,73)$ |
| Order: | \(388944\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 37 \cdot 73 \) |
| Exponent: | \(194472\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 37 \cdot 73 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGL(2,73)$, of order \(388944\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 37 \cdot 73 \) |
| $\operatorname{Aut}(H)$ | $\PGL(2,73)$, of order \(388944\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 37 \cdot 73 \) |
| $W$ | $\PGL(2,73)$, of order \(388944\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 37 \cdot 73 \) |
Related subgroups
| Centralizer: | $C_1$ | |||
| Normalizer: | $\PGL(2,73)$ | |||
| Complements: | $C_2$ | |||
| Minimal over-subgroups: | $\PGL(2,73)$ | |||
| Maximal under-subgroups: | $C_{73}:C_{36}$ | $D_{37}$ | $D_{36}$ | $S_4$ |
Other information
| Möbius function | $-1$ |
| Projective image | $\PGL(2,73)$ |