Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C2×C42 |
| direct product | C3×#16650 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(5,7)(9,12)(10,14)(16,17,20,21)(18,22,23,19)(25,28)(26,30);(9,12)(13,15)(17,21)(18,23)(24,25,27,28)(26,29,30,31);(8,9,11,12)(10,13,14,15)(16,18)(17,22)(19,21)(20,23)(24,25)(26,31)(27,28)(29,30);(4,5)(6,7)(8,10,11,14)(9,13,12,15)(16,19,20,22)(17,23,21,18)(24,26)(25,29)(27,30)(28,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9638);