Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#66677 |
| Hall | C3×C5×C7 ⋊ #41 |
| Hall | C5×C7 ⋊ #235 |
| Hall | C3×C7 ⋊ C5×#41 |
| Hall | C3×C5 ⋊ #1161 |
| Sylow | C7 ⋊ C5×#235 |
| Sylow | C3 ⋊ C5×#1161 |
160 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 64 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)(16,18,19,23)(17,21,20,22);(14,15)(16,17,19,20)(18,22,23,21);(7,8)(9,10)(11,12)(17,20)(21,22)225 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 687);