Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#520⋊C3 |
| unambiguous semidirect product | #520⋊(C3×C7) |
| Fitting, Hall | C7×#520 ⋊ C3 |
| Sylow | #520 ⋊ C3×C7 |
168 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11)(13,17)(14,15,20)(18,25,26,24,19,27)(21,23,22);(12,26)(13,22)(14,25)(15,18)(16,27)(17,20)(19,21)(23,24)266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10036);