Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19063 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3)(6,12)(7,14)(9,17)(10,15)(13,16)(18,19)(20,21)(22,32)(23,37)(24,29)(25,30)(26,31)(27,35)(28,39)(33,41)(34,42)(36,45)(38,43)(40,44)(46,48)(47,50)(49,51);(4,5,8,11)(6,9,13,18)(7,15,14,10)(12,19,16,17)(20,22)(21,27)(23,33)(24,34)(25,35)(26,36)(28,41)(29,32)(30,45)(31,42)(37,50)(38,47)(39,48)(40,49)(43,51)(44,46);(4,6)(5,9)(7,16)(8,13)(10,19)(11,18)(12,14)(15,17)(20,23,26,40)(21,28,31,43)(22,33,36,49)(24,39,25,38)(27,41,42,51)(29,44,30,37)(32,46,45,50)(34,48,35,47);(20,24,26,25)(21,29,31,30)(22,34,36,35)(23,38,40,39)(27,32,42,45)(28,37,43,44)(33,47,49,48)(41,50,51,46)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12051);