Subgroup ($H$) information
| Description: | $\SL(2,79)$ |
| Order: | \(492960\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
| Index: | \(2\) |
| Exponent: | \(246480\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
| Generators: |
$\left(\begin{array}{rr}
10 & 58 \\
65 & 69
\end{array}\right), \left(\begin{array}{rr}
23 & 9 \\
28 & 11
\end{array}\right), \left(\begin{array}{rr}
23 & 15 \\
73 & 3
\end{array}\right), \left(\begin{array}{rr}
39 & 78 \\
20 & 38
\end{array}\right), \left(\begin{array}{rr}
78 & 0 \\
0 & 78
\end{array}\right)$
|
| Derived length: | $0$ |
The subgroup is the commutator subgroup (hence characteristic and normal), maximal, a semidirect factor, nonabelian, and quasisimple (hence nonsolvable and perfect). Whether it is almost simple has not been computed.
Ambient group ($G$) information
| Description: | $\SL(2,79):C_2$ |
| Order: | \(985920\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
| Exponent: | \(492960\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
| Derived length: | $1$ |
The ambient group is nonabelian 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \PSL(2,79).C_2$ |
| $\operatorname{Aut}(H)$ | $\PGL(2,79)$, of order \(492960\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
| $W$ | $\PGL(2,79)$, of order \(492960\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
Related subgroups
| Centralizer: | $C_2$ | ||||
| Normalizer: | $\SL(2,79):C_2$ | ||||
| Complements: | $C_2$ | ||||
| Minimal over-subgroups: | $\SL(2,79):C_2$ | ||||
| Maximal under-subgroups: | $C_{158}:C_{39}$ | $C_5:Q_{32}$ | $C_{39}:C_4$ | $\SL(2,5)$ | $C_2.S_4$ |
Other information
| Möbius function | $-1$ |
| Projective image | $\PGL(2,79)$ |