Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19092 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3219 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
816 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)(5,10)(6,13)(7,8)(12,15)(14,16)(17,19)(20,21)(22,30)(23,27)(24,36)(25,28)(26,33)(29,39)(31,47)(32,44)(34,45)(35,42)(37,41)(38,43)(40,49)(46,48)(50,51);(4,5)(6,11)(7,15)(8,10)(9,12)(13,19)(14,18)(16,17)(20,22,25,32)(21,26,28,41)(23,31,35,48)(24,37,39,33)(27,40,42,51)(29,44,36,30)(34,50,38,49)(43,46,45,47);(4,6,9,14)(5,11,12,18)(7,13,8,16)(10,17,15,19)(20,23,25,35)(21,27,28,42)(22,31,32,48)(24,38,39,34)(26,40,41,51)(29,45,36,43)(30,46,44,47)(33,50,37,49);(5,12)(10,15)(11,18)(17,19)(21,28)(22,32)(27,42)(29,36)(31,48)(33,37)(43,45)(49,50)93 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12080);