Finite Group DatabasebetaFinite Group Databasebeta| order | 420 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C22×C4×AGL(1,𝔽7) (order 672) · Inn(G) = D7 · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×C5×D7 |
| Fitting, socle | C2×C3×C5×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C3×C5 |
| Hall | C3×C5×C7 ⋊ C22 |
| Hall | C5×C7 ⋊ C22×C3 |
| Hall | C3×C7 ⋊ C22×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C5 | D7 |
| commutator, cosocle, hypocenter | C7 | C22×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3×C5 | |
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)(12,13)(14,15)(16,17);(1,2)(11,12,14,16,17,15,13)150 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D7×Dic5, C3×#63305, C3×#63306, … |
| maximal normal subgroup of | C3×D7×Dic5, C3×#63305, C3×#63306, … |
| maximal quotient of | C3×C5×Dic14, C3×C4×C5×D7, C3×C5×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(420, 34);