Finite Group DatabasebetaFinite Group Databasebeta| order | 420 |
| isoclinism | isoclinic class of D105 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D210 |
| direct product | C2×D105 |
| unambiguous semidirect product | (C2×C3)⋊D35 |
| unambiguous semidirect product | (C2×C5)⋊D21 |
| unambiguous semidirect product | (C2×C7)⋊D15 |
| Fitting, socle | C2×C3×C5×C7 ⋊ C2 |
| derived, Hall | C3×C5×C7 ⋊ C22 |
| Hall | C5×C7 ⋊ C2×D3 |
| Hall | C3×C7 ⋊ C2×D5 |
| Hall | C3×C5 ⋊ C2×D7 |
| Sylow | C7 ⋊ C2×D15 |
| Sylow | C5 ⋊ C2×D21 |
| Sylow | C3 ⋊ C2×D35 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D105 |
| commutator, cosocle, hypocenter | C3×C5×C7 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5×C7 | C2 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 210 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2)(3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17);(4,5)(7,10)(8,9)(12,17)(13,16)(14,15)108 conjugacy classes — view the full character table.
| maximal subgroup of | #11719756, #11719758, #11719759, … |
| maximal normal subgroup of | #11719756, #11719758, #11719759, … |
| maximal quotient of | Dic210, C4×D105, D420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(420, 40);