Subgroup ($H$) information
| Description: | $\PSL(2,59)$ |
| Order: | \(102660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 29 \cdot 59 \) |
| Index: | $1$ |
| Exponent: | \(51330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 29 \cdot 59 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
22 & 10 \\
40 & 37
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
43 & 34 \\
12 & 15
\end{array}\right) \right]$
|
| Derived length: | $0$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the socle, a direct factor, nonabelian, a Hall subgroup, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
Ambient group ($G$) information
| Description: | $\PSL(2,59)$ |
| Order: | \(102660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 29 \cdot 59 \) |
| Exponent: | \(51330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 29 \cdot 59 \) |
| Derived length: | $0$ |
The ambient group is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGL(2,59)$, of order \(205320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 29 \cdot 59 \) |
| $\operatorname{Aut}(H)$ | $\PGL(2,59)$, of order \(205320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 29 \cdot 59 \) |
| $W$ | $\PSL(2,59)$, of order \(102660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 29 \cdot 59 \) |
Related subgroups
| Centralizer: | $C_1$ | ||||
| Normalizer: | $\PSL(2,59)$ | ||||
| Complements: | $C_1$ | ||||
| Maximal under-subgroups: | $C_{59}:C_{29}$ | $A_5$ | $A_5$ | $D_{30}$ | $D_{29}$ |
Other information
| Möbius function | $1$ |
| Projective image | $\PSL(2,59)$ |