Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#18465 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#538 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3)(7,14)(9,15)(10,17)(12,18)(20,21,26,31)(22,32,35,45)(23,28,37,43)(24,30,25,29)(27,39,41,34)(33,46,49,50)(36,44,38,42)(40,47,51,48);(4,5)(6,9)(7,10)(8,11)(12,16)(13,15)(14,17)(18,19)(20,22,26,35)(21,27,31,41)(23,33,37,49)(24,34,25,39)(28,40,43,51)(29,32,30,45)(36,47,38,48)(42,50,44,46);(4,6)(5,9)(7,12)(8,13)(10,16)(11,15)(14,18)(17,19)(20,23)(21,28)(22,33)(24,38)(25,36)(26,37)(27,40)(29,44)(30,42)(31,43)(32,46)(34,48)(35,49)(39,47)(41,51)(45,50);(4,7,8,14)(5,10,11,17)(6,12,13,18)(9,16,15,19)(20,24,26,25)(21,29,31,30)(22,34,35,39)(23,36,37,38)(27,32,41,45)(28,42,43,44)(33,47,49,48)(40,50,51,46)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11453);