Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of D3×D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×D21⋊C4 |
| unambiguous semidirect product | D21⋊(C4×C5) |
| derived, Hall | C3×C7 ⋊ C2×C4×C5 |
| Hall | C3×C5×C7 ⋊ C2×C4 |
| Hall | C5×C7 ⋊ C4×D3 |
| Hall | C3×C5 ⋊ C4×D7 |
| Sylow | C7 ⋊ C4×C5×D3 |
| Sylow | C3 ⋊ C4×C5×D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D3×D7 |
| commutator, hypocenter | C3×C7 | C2×C4×C5 |
| Frattini | C2 | C5×D3×D7 |
| Fitting, socle | C2×C3×C5×C7 | C22 |
| prime radical | C2×C5×D21 | C2 |
| cosocle | C2×C3×C7 | C22×C5 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×C5 | |
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)(6,7,9,11,12,10,8)(14,15)(16,18,19,17);(7,8)(9,10)(11,12)(13,14,15)(16,17,19,18)150 conjugacy classes — view the full character table.
| maximal subgroup of | D5×D21⋊C4, #13132486, #13132503, … |
| maximal normal subgroup of | D5×D21⋊C4, #13132486, #13132503, … |
| maximal quotient of | C5×D21⋊C8, C5×#66559, C5×Dic3×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 51);