Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×D21⋊Q8 |
| unambiguous semidirect product | D21⋊(C5×Q8) |
| Hall | C3×C5×C7 ⋊ C2×Q8 |
| Hall | C5×C7 ⋊ D3×Q8 |
| Hall | C3×C7 ⋊ C2×C5×Q8 |
| Hall | C3×C5 ⋊ Q8×D7 |
| Sylow | C7 ⋊ C5×D3×Q8 |
| Sylow | C3 ⋊ C5×Q8×D7 |
152 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 152 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) = 23 |
| transitive degree | μt(G) = 840 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5)(6,7,9,11,12,10,8)(13,14,15)(16,19,20,23)(17,21,18,22);(14,15)(16,18,20,17)(19,21,23,22);(7,8)(9,10)(11,12)(16,17,20,18)(19,22,23,21)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 684);