Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#4474 |
| Fitting | C3×C7×#99 ⋊ C2 |
| Hall | C3×C7 ⋊ #353 |
| Sylow | C3 ⋊ C7×#353 |
172 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5,6,7)(8,9,10)(11,12)(13,15,17,18)(14,16);(9,10)(12,15)(13,17)(16,18)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 545);