Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#4726 |
| Fitting | C3×C7×C22⋊C8 ⋊ C2 |
| Hall | C3×C7 ⋊ #407 |
| Sylow | C3 ⋊ C7×#407 |
380 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3,4,5,6,7)(8,9,10)(12,14)(15,16,17,20,18,19,21,22);(11,12)(13,14);(9,10)(16,19)(20,22)336 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 5051);