Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#15062 |
| 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 | |
1106 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)(5,6)(9,12)(13,15)(16,17,20,19)(18,22,23,21)(24,25,28,29)(26,30,31,27);(4,5,7,6)(18,23)(21,22)(25,29)(26,31);(8,9,11,12)(10,13,14,15)(16,18,20,23)(17,21,19,22)(24,26)(25,30)(27,29)(28,31);(4,6,7,5)(8,10,11,14)(9,13,12,15)(16,19,20,17)(18,22,23,21)(24,27,28,30)(25,26,29,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8050);