Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#41418 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(5,7)(8,9,13,14)(10,15,17,12)(11,19,21,16)(18,20,23,22)(24,25,29,33)(26,30,34,39)(27,31,37,36)(28,32,38,35);(9,14)(10,17)(16,19)(18,23)(26,34)(28,38)(30,39)(32,35);(4,5,6,7)(8,10)(9,15)(11,18)(12,14)(13,17)(16,22)(19,20)(21,23)(24,26)(25,30)(27,32)(28,36)(29,34)(31,38)(33,39)(35,37);(8,11,13,21)(9,16,14,19)(10,18,17,23)(12,20,15,22)(24,27)(25,31)(26,35)(28,30)(29,37)(32,34)(33,36)(38,39);(8,12,13,15)(9,10,14,17)(11,20,21,22)(16,18,19,23)(24,28)(25,32)(26,36)(27,30)(29,38)(31,34)(33,35)(37,39)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34406);