Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of #63305 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#63307 |
| Hall | C3×C5×C7 ⋊ D4 |
| Hall | C5×C7 ⋊ C3×D4 |
| Hall | C3×C7 ⋊ D20 |
| Hall | C3×C5 ⋊ C22⋊D7 |
| Sylow | C7 ⋊ C3×D20 |
| Sylow | C5 ⋊ C3×C22⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D5×D7 |
| commutator, cosocle | C2×C5×C7 | C22×C3 |
| Frattini | C2 | C3×D5×D7 |
| Fitting, socle | C2×C3×C5×C7 | C22 |
| hypocenter | C5×C7 | C3×D4 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2×C3 | |
64 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)(4,5,7,9,10,8,6)(12,13)(14,15)(17,19);(5,6)(7,8)(9,10)(11,12,14,15,13)(16,17,18,19)111 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63307, #13132503, #13132506, … |
| maximal normal subgroup of | D3×#63307, #13132503, #13132506, … |
| maximal quotient of | C3×#10587843, C3×#10587848, C3×#10587849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 47);