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×C4 |
| direct product | C3×#16716 |
| 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 | |
1086 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)(4,5)(6,7)(9,12)(10,14)(16,17,19,21)(18,22,23,20)(24,25)(26,30)(27,31)(28,29);(4,5,7,6)(18,23)(20,22)(25,29)(26,31);(5,6)(8,9,11,12)(10,13,14,15)(16,18,19,23)(17,20,21,22)(24,26,28,31)(25,30,29,27);(4,6,7,5)(8,10)(9,13)(11,14)(12,15)(16,17)(18,20)(19,21)(22,23)(24,27,28,30)(25,26,29,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9704);