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×#62978 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6326 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)(20,21,27,24)(22,31,23,28)(25,29,39,36)(26,37,40,30)(32,43,34,41)(33,42,35,45)(38,48,49,44)(46,50,47,51);(6,13)(7,15)(10,12)(11,16)(20,22,27,23)(21,28,24,31)(25,34,39,32)(26,33,40,35)(29,43,36,41)(30,42,37,45)(38,47,49,46)(44,51,48,50);(5,9)(10,12)(11,16)(17,18)(20,23,27,22)(21,28,24,31)(25,34,39,32)(26,35,40,33)(29,41,36,43)(30,42,37,45)(38,47,49,46)(44,50,48,51);(4,6)(5,10)(7,14)(8,13)(9,12)(11,17)(15,19)(16,18)(20,24,27,21)(22,31,23,28)(25,36,39,29)(26,37,40,30)(32,43,34,41)(33,45,35,42)(38,48,49,44)(46,51,47,50);(4,5)(6,10)(7,11)(8,9)(12,13)(14,17)(15,16)(18,19)(20,25)(21,29)(22,32)(23,34)(24,36)(26,38)(27,39)(28,41)(30,44)(31,43)(33,46)(35,47)(37,48)(40,49)(42,50)(45,51);(4,7)(5,11)(6,14)(8,15)(9,16)(10,17)(12,18)(13,19)(20,26)(21,30)(22,33)(23,35)(24,37)(25,38)(27,40)(28,42)(29,44)(31,45)(32,46)(34,47)(36,48)(39,49)(41,50)(43,51)201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55966);