Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#4966 |
| Fitting | C2×C3×C7×C4⋊C4 ⋊ C2 |
| Hall | C3×C7 ⋊ #417 |
| Sylow | C3 ⋊ C7×#417 |
324 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 672 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3,4,5,6,7)(8,9,10)(11,12,13,14)(16,21)(17,18)(20,22);(15,17,19,18)(16,20,22,21);(9,10)(15,16)(17,20)(18,21)(19,22)294 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 5309);