Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | ΓL(1,𝔽8)×#709 |
| unambiguous semidirect product | (C7×#709)⋊C3 |
| Fitting | C2×C4×C5×C7 ⋊ C2×C3 |
| Hall | C7×#709 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ #41 |
| Hall | C5×C7 ⋊ C3×#41 |
| Sylow | C7 ⋊ C3×#709 |
| Sylow | C5 ⋊ #41×ΓL(1,𝔽8) |
160 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3,4)(5,7,6)(8,9,11,12,10)(14,18)(17,20);(1,2,3,5,4,6,7)(13,15,16,19)(14,17,18,20);(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)130 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#41×ΓL(1,𝔽8), ΓL(1,𝔽8)×#4054, ΓL(1,𝔽8)×#4078, … |
| maximal normal subgroup of | D5×#41×ΓL(1,𝔽8), ΓL(1,𝔽8)×#4054, ΓL(1,𝔽8)×#4078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 588);