Finite Group DatabasebetaFinite Group Databasebeta| order | 420 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C23×C3×AGL(1,𝔽5) (order 480) · Inn(G) = D5 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C3×C7×Dic5 |
| derived, Sylow | C5 ⋊ C3×C4×C7 |
| Hall | C3×C5×C7 ⋊ C4 |
| Hall | C5×C7 ⋊ C3×C4 |
| Hall | C3×C5 ⋊ C4×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C7 | D5 |
| commutator, hypocenter | C5 | C3×C4×C7 |
| Frattini | C2 | C3×C7×D5 |
| Fitting, socle, prime radical | C2×C3×C5×C7 | C2 |
| cosocle | C2×C5 | C2×C3×C7 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C3×C7 | |
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)(6,7,8,9)(10,11,12,13,14,15,16)(17,18,19);(1,2,4,5,3)168 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C7×#170, C3×C7⋊#170, C7×C3⋊#170, … |
| maximal normal subgroup of | C3×C7×#170, C3×C7⋊#170, C7×C3⋊#170, … |
| maximal quotient of | C3×C7×#168, C2×C3×C7×Dic5, C7×C9×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(420, 6);