Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#89760 |
| Fitting | C2×C3×C7×C4⋊C4 ⋊ C2 |
| Hall | C3×C7 ⋊ #390 |
| Sylow | C7 ⋊ C3×#390 |
516 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 166 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)(4,5,7,9,10,8,6)(11,12,13,14)(16,18)(20,22);(19,20,21,22);(5,6)(7,8)(9,10)(15,16)(17,18)(20,22)264 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4712);