Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#10872 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #121 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #380 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(8,9,13,17,14,18,12,28)(10,19,21,15,23,30,22,31)(11,24,26,32,27,38,25,16)(20,29,34,37,36,39,35,33);(4,5,6,7)(8,10)(9,15)(11,20)(12,22)(13,21)(14,23)(16,29)(17,30)(18,31)(19,28)(24,37)(25,35)(26,34)(27,36)(32,39)(33,38);(8,11,12,25,14,27,13,26)(9,16,17,24,18,32,28,38)(10,20,22,35,23,36,21,34)(15,29,30,37,31,39,19,33)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3860);