Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#35311 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(5,7)(10,14)(12,15)(16,17,20,23)(18,21,26,24)(19,27,28,22)(25,30,31,29);(4,5)(6,7)(8,9,11,13)(10,12,14,15)(16,17,20,23)(18,24,26,21)(19,22,28,27)(25,30,31,29);(8,10)(9,12)(11,14)(13,15)(16,17)(18,21)(19,22)(20,23)(24,26)(25,29)(27,28)(30,31);(16,18)(17,21)(19,25)(20,26)(22,29)(23,24)(27,30)(28,31);(16,19,20,28)(17,22,23,27)(18,25,26,31)(21,29,24,30)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28299);