Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
810 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)(4,5)(6,13)(7,11)(8,16)(9,12)(10,14)(15,18)(17,19)(20,21,26,29)(22,32,35,44)(23,36,38,45)(24,31,25,30)(27,39,42,34)(28,40,43,37)(33,41,49,51)(46,47,50,48);(4,5)(6,10)(7,11)(8,16)(9,12)(13,17)(14,18)(15,19)(20,22,26,35)(21,27,29,42)(23,33,38,49)(24,39,25,34)(28,41,43,51)(30,44,31,32)(36,50,45,46)(37,48,40,47);(4,6)(5,10)(7,14)(8,17)(9,15)(11,18)(12,19)(13,16)(20,23)(21,28)(22,33)(24,37)(25,40)(26,38)(27,41)(29,43)(30,36)(31,45)(32,46)(34,47)(35,49)(39,48)(42,51)(44,50);(4,7,9,8)(5,11,12,16)(6,14,15,17)(10,18,19,13)(21,29)(27,42)(28,43)(30,31)(32,44)(36,45)(41,51)(46,50)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12001);