Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#89290 |
| Fitting | C3×C7×#108 ⋊ C2 |
| Hall | C3×C7 ⋊ #361 |
| Sylow | C7 ⋊ C3×#361 |
124 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3)(4,5,7,9,10,8,6)(11,12,13,18,16,19,15,24)(14,22,20,17,23,25,21,26);(5,6)(7,8)(9,10)(12,17)(13,15)(14,21)(18,22)(19,26)(20,23)(24,25)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 444);