Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#16695 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(5,7)(8,9)(10,12)(11,13)(14,15)(16,17,19,21)(18,22,23,20)(25,27)(29,30);(9,10)(13,14)(16,17,19,21)(18,20,23,22)(24,25,28,27)(26,30,31,29);(4,5)(6,7)(8,10,12,9)(11,13,15,14)(16,17)(18,20)(19,21)(22,23)(24,26,28,31)(25,29,27,30);(8,11,12,15)(9,13,10,14)(16,18)(17,20)(19,23)(21,22)(24,27,28,25)(26,29,31,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9683);