Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#5779 |
| Fitting | C3×C7×#137 ⋊ C2 |
| Hall | C3×C7 ⋊ #578 |
| Sylow | C3 ⋊ C7×#578 |
496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2,3,4,5,6,7)(8,9,10)(11,14,16,24)(12,19,17,31)(13,21,23,39)(15,28,29,26)(18,33,30,41)(20,38,32,36)(22,40,27,25)(34,42,37,35);(11,13,16,23)(12,18,17,30)(14,25,24,40)(15,27,29,22)(19,35,31,42)(20,37,32,34)(21,28,39,26)(33,38,41,36);(12,17)(14,24)(18,30)(20,32)(21,39)(25,40)(26,28)(34,37);(9,10)(11,12)(13,18)(14,19)(15,20)(16,17)(21,33)(22,34)(23,30)(24,31)(25,35)(26,36)(27,37)(28,38)(29,32)(39,41)(40,42)231 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10784);