Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19651 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#19667 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(9,13)(11,14)(15,19)(16,20)(18,21)(22,23)(24,25)(26,33)(27,31)(28,37)(29,32)(30,35)(34,38)(36,39);(8,9,14,20,12,16,11,13)(10,15,21,23,17,22,18,19);(8,10,12,17)(9,15,16,22)(11,18,14,21)(13,19,20,23)(24,26,28,35,29,36,27,34)(25,30,31,33,32,38,37,39);(10,17)(15,22)(18,21)(19,23)(24,27,29,28)(25,31,32,37)(26,34,36,35)(30,33,38,39)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12655);