Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#60612 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C23×D4 |
| prime radical | C3×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×#117 ◁ G | |
2050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,24)(15,29)(16,21)(17,22)(18,23)(19,27)(20,31)(25,33)(26,35)(28,34)(30,37)(32,36)(38,41)(39,43)(40,42);(5,8)(9,11)(12,14,17,28,18,26,16,27)(13,19,21,35,23,34,22,24)(15,25,31,41,32,39,30,40)(20,33,29,42,37,43,36,38);(4,6,7,10)(5,9,8,11)(12,15,16,30,18,32,17,31)(13,20,22,36,23,37,21,29)(14,25,27,40,26,39,28,41)(19,33,24,38,34,43,35,42);(14,26)(19,34)(24,35)(25,39)(27,28)(33,43)(38,42)(40,41);(12,16,18,17)(13,21,23,22)(14,27,26,28)(15,30,32,31)(19,35,34,24)(20,29,37,36)(25,40,39,41)(33,42,43,38)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53600);