Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C2×C42 |
| direct product | C3×#14909 |
| 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 | |
1130 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)(10,13,14,15)(16,17,20,21)(18,19,23,22)(24,25)(26,27)(28,30)(29,31);(10,14)(13,15)(17,21)(18,23)(24,26,28,31)(25,27,30,29);(8,10)(9,13)(11,15)(12,14)(16,18,20,23)(17,22,21,19)(24,26)(25,29)(27,30)(28,31);(4,5)(6,7)(8,11,12,9)(10,15,14,13)(16,19,20,22)(17,23,21,18)(24,27,28,29)(25,26,30,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 7897);