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×#25751 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1474 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)(10,19)(11,23)(12,17)(13,27)(14,18)(15,21)(16,26)(20,37)(22,31)(24,30)(25,35)(28,32)(29,34)(33,38)(36,39);(4,5,7,6)(8,10,12,21)(9,15,17,19)(11,24,25,39)(13,22,14,28)(16,33,34,37)(18,31,27,32)(20,29,38,26)(23,36,35,30);(8,11,13,26,12,25,14,29)(9,16,18,23,17,34,27,35)(10,20,22,39,21,38,28,24)(15,30,31,33,19,36,32,37);(4,6,7,5)(9,17)(11,25)(15,19)(18,27)(20,38)(24,39)(26,29)(31,32)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18739);