Finite Group DatabasebetaFinite Group Databasebeta| order | 420 |
| isoclinism | isoclinic class of D21 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×Dic21 |
| unambiguous semidirect product | C3⋊(C5×Dic7) |
| unambiguous semidirect product | C7⋊(C5×Dic3) |
| derived, Hall | C3×C7 ⋊ C4×C5 |
| Hall | C3×C5×C7 ⋊ C4 |
| Hall | C5×C7 ⋊ Dic3 |
| Hall | C3×C5 ⋊ Dic7 |
| Sylow | C7 ⋊ C5×Dic3 |
| Sylow | C3 ⋊ C5×Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D21 |
| commutator, hypocenter | C3×C7 | C4×C5 |
| Frattini | C2 | C5×D21 |
| Fitting, socle, prime radical | C2×C3×C5×C7 | C2 |
| cosocle | C2×C3×C7 | C2×C5 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×C5 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 420 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,3)(4,5)(6,7)(9,10)(11,12,13,14)(15,16,17,18,19);(1,2,4,6,7,5,3)(8,9,10)120 conjugacy classes — view the full character table.
| maximal subgroup of | C5×Dic3×D7, C5×D3×Dic7, C5×#4245, … |
| maximal normal subgroup of | C5×Dic3×D7, C5×D3×Dic7, C5×#4245, … |
| maximal quotient of | C5×C3⋊(C7⋊C8), C2×C5×Dic21, C5×Dic63, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(420, 9);