Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25445 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3060 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
886 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)(8,9,12,15,14,19,13,18)(10,20,21,16,24,31,23,32)(11,25,27,17,26,29,28,33)(22,30,35,38,37,39,36,34);(4,5)(6,7)(9,15,19,18)(10,21,24,23)(11,26)(17,25,33,29)(22,36,37,35)(27,28)(30,39)(34,38);(5,7)(8,10)(9,16)(11,22)(12,23)(13,21)(14,24)(15,20)(17,30)(18,31)(19,32)(25,38)(26,37)(27,36)(28,35)(29,34)(33,39);(8,11,13,28,14,26,12,27)(9,17,15,29,19,33,18,25)(10,22,21,35,24,37,23,36)(16,30,20,34,32,39,31,38)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18433);