Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#25750 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(5,6)(8,9,14,18)(10,15,21,32)(11,22,24,35)(12,26,13,17)(16,28,34,25)(19,33,36,39)(20,31,27,30)(23,38,37,29);(4,5,7,6)(8,10)(9,15)(11,23)(12,20)(13,27)(14,21)(16,33)(17,30)(18,32)(19,25)(22,38)(24,37)(26,31)(28,36)(29,35)(34,39);(8,11,12,25,14,24,13,28)(9,16,17,22,18,34,26,35)(10,19,20,37,21,36,27,23)(15,29,30,33,32,38,31,39);(4,6,7,5)(11,24)(16,34)(19,36)(22,35)(23,37)(25,28)(29,38)(33,39)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18738);