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) = C22×C3×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×C7×D5 |
| unambiguous semidirect product | C5⋊(C22×C3×C7) |
| unambiguous semidirect product | (C2×C5)⋊(C2×C3×C7) |
| Fitting, socle | C2×C3×C5×C7 ⋊ C2 |
| derived, Sylow | C5 ⋊ C22×C3×C7 |
| Hall | C3×C5×C7 ⋊ C22 |
| Hall | C5×C7 ⋊ C22×C3 |
| Hall | C3×C5 ⋊ C22×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C7 | D5 |
| commutator, cosocle, hypocenter | C5 | C22×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5×C7 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C3×C7 | |
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)(7,8)(9,10)(11,12,13,14,15,16,17);(1,2)(6,7,9,10,8)168 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5×Dic7, C3×#63305, C3×#63307, … |
| maximal normal subgroup of | C3×D5×Dic7, C3×#63305, C3×#63307, … |
| maximal quotient of | C3×C7×Dic10, C3×C4×C7×D5, C3×C7×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(420, 35);