Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#66681 |
| Fitting | C3×C4×C5×C7 ⋊ C22 |
| Hall | C3×C5×C7 ⋊ C2×D4 |
| Hall | C5×C7 ⋊ D3×D4 |
| Hall | C3×C7 ⋊ C2×C5×D4 |
| Hall | C3×C5 ⋊ D4×D7 |
| Sylow | C7 ⋊ C5×D3×D4 |
| Sylow | C3 ⋊ C5×D4×D7 |
216 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 420 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,4,5)(6,7,9,11,12,10,8)(13,14,15)(16,17,18,19);(14,15)(17,19);(7,8)(9,10)(11,12)(17,19)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 691);