Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of #1201 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#6667 |
| Hall | C3×C5×C7 ⋊ C22⋊C4 |
| Hall | C5×C7 ⋊ (C2×D3)⋊C4 |
| Hall | C3×C7 ⋊ #685 |
| Hall | C3×C5 ⋊ C7×C22⋊C4 |
| Sylow | C5 ⋊ C7×(C2×D3)⋊C4 |
| Sylow | C3 ⋊ C7×#685 |
136 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 2 generators on 23 points:
(1,2,3,4,5,6,7)(8,9,11,12,10)(14,15)(16,18,19,17)(21,23);(9,10)(11,12)(13,14,15)(16,17,19,18)(20,21,22,23)294 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 188);