Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#4734 |
| Fitting | C3×C7×C22⋊C8 ⋊ C2 |
| Hall | C3×C7 ⋊ #449 |
| Sylow | C3 ⋊ C7×#449 |
324 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,3,4,5,6,7)(8,9,10)(12,14)(15,18,17,16,20,22,19,21);(11,12)(13,14)(15,17,20,19)(16,18,21,22);(9,10)(15,16)(17,18)(19,22)(20,21)273 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 5058);