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×#19060 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3553 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
822 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)(20,21,26,31)(22,32,35,44)(23,36,37,45)(24,30,25,29)(27,39,42,34)(28,40,43,38)(33,41,47,51)(46,48,50,49);(4,5,8,11)(6,9,13,18)(7,15,14,10)(12,19,16,17)(20,22)(21,27)(23,33)(24,39)(25,34)(26,35)(28,41)(29,44)(30,32)(31,42)(36,50)(37,47)(38,49)(40,48)(43,51)(45,46);(4,6)(5,9)(7,16)(8,13)(10,19)(11,18)(12,14)(15,17)(20,23)(21,28)(22,33)(24,38)(25,40)(26,37)(27,41)(29,36)(30,45)(31,43)(32,46)(34,48)(35,47)(39,49)(42,51)(44,50);(23,37)(28,43)(33,47)(36,45)(38,40)(41,51)(46,50)(48,49)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12048);