Subgroup ($H$) information
| Description: | $\SL(2,79)$ |
| Order: | \(492960\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
| Index: | \(13\) |
| Exponent: | \(246480\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
| Generators: |
$\left(\begin{array}{rr}
28 & 22 \\
11 & 51
\end{array}\right), \left(\begin{array}{rr}
4 & 75 \\
33 & 66
\end{array}\right), \left(\begin{array}{rr}
78 & 0 \\
0 & 78
\end{array}\right)$
|
| Derived length: | $0$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and quasisimple (hence nonsolvable and perfect). Whether it is a direct factor, a semidirect factor, or almost simple has not been computed.
Ambient group ($G$) information
| Description: | $C_{13}.\SL(2,79)$ |
| Order: | \(6408480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 13^{2} \cdot 79 \) |
| Exponent: | \(246480\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.
Quotient group ($Q$) structure
| Description: | $C_{13}$ |
| Order: | \(13\) |
| Exponent: | \(13\) |
| Automorphism Group: | $C_{12}$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_{12}$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{12}\times \PSL(2,79).C_2$ |
| $\operatorname{Aut}(H)$ | $\PGL(2,79)$, of order \(492960\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |