Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of #6758 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×C4⋊ΓL(1,𝔽16) |
| unambiguous semidirect product | C4⋊(C7×ΓL(1,𝔽16)) |
| Fitting | C3×C4×C5×C7 ⋊ C4 |
| Hall | C3×C5×C7 ⋊ C4⋊C4 |
| Hall | C5×C7 ⋊ C4⋊Dic3 |
| Hall | C3×C7 ⋊ C4⋊AGL(1,𝔽5) |
| Hall | C3×C5 ⋊ C7×C4⋊C4 |
| Sylow | C5 ⋊ C7×C4⋊Dic3 |
| Sylow | C3 ⋊ C7×C4⋊AGL(1,𝔽5) |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 420 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,4,5,6,7)(8,9,12,10,11)(13,14,15)(16,17,18,19);(9,10,11,12)(14,15)(17,19)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 640);