Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#89270 |
| Fitting | C3×C7×#100 ⋊ C2 |
| Hall | C3×C7 ⋊ AΣL(1,𝔽16) |
| Sylow | C7 ⋊ C3×AΣL(1,𝔽16) |
188 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 188 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)(4,5,7,9,10,8,6)(11,12,13,16,14,17,18,15);(5,6)(7,8)(9,10)(12,15)(16,17)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 424);