Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62972 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62977 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 51 points:
(1,2,3)(4,5)(6,12)(7,11)(8,9)(10,13)(14,18)(15,16)(17,19)(20,21)(22,30)(23,26)(24,28)(25,35)(27,33)(29,39)(31,42)(32,41)(34,36)(37,40)(38,47)(43,48)(44,46)(45,50)(49,51);(6,13)(7,15)(10,12)(11,16)(21,23)(24,36)(25,39)(27,30)(31,37)(32,46)(43,47)(50,51);(5,9)(10,12)(11,16)(17,18)(20,22,26,33)(21,27,23,30)(24,37,36,31)(25,32,39,46)(28,42,34,40)(29,41,35,44)(38,49,48,45)(43,51,47,50);(4,6)(5,10)(7,14)(8,13)(9,12)(11,17)(15,19)(16,18)(20,23,26,21)(22,27,33,30)(24,34,36,28)(25,35,39,29)(31,40,37,42)(32,41,46,44)(38,47,48,43)(45,50,49,51);(4,5)(6,10)(7,11)(8,9)(12,13)(14,17)(15,16)(18,19)(20,24)(21,28)(22,31)(23,34)(25,38)(26,36)(27,40)(29,43)(30,42)(32,45)(33,37)(35,47)(39,48)(41,50)(44,51)(46,49);(4,7)(5,11)(6,14)(8,15)(9,16)(10,17)(12,18)(13,19)(20,25)(21,29)(22,32)(23,35)(24,38)(26,39)(27,41)(28,43)(30,44)(31,45)(33,46)(34,47)(36,48)(37,49)(40,50)(42,51)201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55965);