Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#90323 |
| Fitting | C3×C7×C22⋊Q8 ⋊ C2 |
| Hall | C3×C7 ⋊ D4×Q8 |
| Sylow | C7 ⋊ C3×D4×Q8 |
560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(4,5,7,9,10,8,6)(12,14)(15,17,18,22)(16,19,20,21);(15,16,18,20)(17,21,22,19);(11,12)(13,14);(5,6)(7,8)(9,10)(12,14)201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10253);