Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C29 |
| direct product | C7×#60183 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | C22×D4 |
| commutator | C4 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C2×C7 | C22×C4×D4 |
| prime radical | C7×#3584 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C2×C7×#131 ◁ G | |
2612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,18,14,10,19,17)(11,15,23,29,24,20,32,28)(12,16,26,31,25,21,33,30)(22,27,36,38,35,34,39,37);(8,10)(9,14)(11,20)(12,21)(13,17)(15,24)(16,25)(18,19)(22,34)(23,28)(26,30)(27,35)(29,32)(31,33)(36,37)(38,39);(8,11,13,23,14,24,19,32)(9,15,18,29,10,20,17,28)(12,22,26,36,25,35,33,39)(16,27,31,38,21,34,30,37);(8,12)(9,16)(10,21)(11,22)(13,26)(14,25)(15,27)(17,30)(18,31)(19,33)(20,34)(23,36)(24,35)(28,37)(29,38)(32,39);(12,25)(16,21)(22,35)(26,33)(27,34)(30,31)(36,39)(37,38)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53171);