Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63057 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 6 generators on 51 points:
(1,2,3)(5,9)(7,15)(10,12)(13,19)(21,26)(27,37)(28,38)(29,39)(40,48)(41,49)(42,43)(50,51);(4,5,8,9)(6,12,14,10)(7,11,15,16)(13,18,19,17)(21,26)(22,30)(28,38)(29,39)(31,44)(32,45)(42,43)(46,47);(4,6,8,14)(5,10,9,12)(7,13,15,19)(11,17,16,18)(20,21)(22,27)(23,28)(24,29)(25,26)(30,37)(31,40)(32,41)(33,42)(34,38)(35,43)(36,39)(44,48)(45,49)(46,50)(47,51);(4,7)(5,11)(6,13)(8,15)(9,16)(10,17)(12,18)(14,19)(20,22)(21,27)(23,31)(24,32)(25,30)(26,37)(28,40)(29,41)(33,46)(34,44)(35,47)(36,45)(38,48)(39,49)(42,50)(43,51);(20,23,25,34)(21,28,26,38)(22,31,30,44)(24,35,36,33)(27,40,37,48)(29,43,39,42)(32,47,45,46)(41,51,49,50);(20,24,25,36)(21,29,26,39)(22,32,30,45)(23,33,34,35)(27,41,37,49)(28,42,38,43)(31,46,44,47)(40,50,48,51)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56045);