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×#16636 |
| 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 | |
1050 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)(8,9,12,11)(10,14,15,13)(16,17,20,21)(18,19,23,22)(25,27)(26,30);(10,15)(13,14)(17,21)(18,23)(24,25,28,27)(26,29,30,31);(8,10,12,15)(9,13,11,14)(16,18,20,23)(17,22,21,19)(24,26)(25,29)(27,31)(28,30);(4,5)(6,7)(8,11,12,9)(10,14,15,13)(16,19,20,22)(17,18,21,23)(24,27,28,25)(26,29,30,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9624);