Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | ΓL(1,𝔽8)×#711 |
| unambiguous semidirect product | (C7×#711)⋊C3 |
| Fitting | C5×C7×D4 ⋊ C2×C3 |
| Hall | C7×#711 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ #41 |
| Hall | C5×C7 ⋊ C3×#41 |
| Sylow | C7 ⋊ C3×#711 |
| Sylow | C5 ⋊ #41×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D5×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C2×C5×C7 | C23×C3 |
| Fitting | C5×C7×D4 | C2×C3 |
| hypercenter | D4 | D5×ΓL(1,𝔽8) |
| hypocenter | C5×C7 | C3×#41 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ D4 | |
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)(15,19);(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)100 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#41×ΓL(1,𝔽8), ΓL(1,𝔽8)×#4057, #72218243, … |
| maximal normal subgroup of | D5×#41×ΓL(1,𝔽8), ΓL(1,𝔽8)×#4057, #72218243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 590);