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×#41416 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(5,7)(8,9,13,17)(10,14,18,23)(11,15,21,20)(12,16,22,19)(25,26)(27,34)(31,33)(35,39);(10,18)(12,22)(14,23)(16,19)(24,25,29,26)(27,30,34,32)(28,33,36,31)(35,38,39,37);(4,5,6,7)(8,10)(9,14)(11,16)(12,20)(13,18)(15,22)(17,23)(19,21)(24,26,29,25)(27,30,34,32)(28,33,36,31)(35,37,39,38);(8,11)(9,15)(10,19)(12,14)(13,21)(16,18)(17,20)(22,23)(24,27)(25,30)(26,32)(28,35)(29,34)(31,37)(33,38)(36,39);(8,12)(9,16)(10,20)(11,14)(13,22)(15,18)(17,19)(21,23)(24,28,29,36)(25,31,26,33)(27,35,34,39)(30,37,32,38)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34404);