Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C7×#67⋊C8 |
| unambiguous semidirect product | #67⋊(C7×C8) |
| Fitting | C3×C7×C22⋊C8 ⋊ C2 |
| Hall | C3×C7 ⋊ C8×D4 |
| Sylow | C3 ⋊ C7×C8×D4 |
268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3,4,5,6,7)(8,9,10)(12,14)(15,16,17,19,18,20,21,22);(11,12)(13,14);(9,10)(12,14)420 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 5050);