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) = C3×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×Dic7 |
| derived, Sylow | C7 ⋊ C3×C4×C5 |
| Hall | C3×C5×C7 ⋊ C4 |
| Hall | C5×C7 ⋊ C3×C4 |
| Hall | C3×C7 ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C5 | D7 |
| commutator, hypocenter | C7 | C3×C4×C5 |
| Frattini | C2 | C3×C5×D7 |
| Fitting, socle, prime radical | C2×C3×C5×C7 | C2 |
| cosocle | C2×C7 | C2×C3×C5 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3×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)(8,11,10,9)(12,13,14,15,16)(17,18,19);(1,2,4,6,7,5,3)150 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5×Dic7, C3×#63304, C3×#63307, … |
| maximal normal subgroup of | C3×D5×Dic7, C3×#63304, C3×#63307, … |
| maximal quotient of | C3×C5×C7⋊C8, C2×C3×C5×Dic7, C5×C9×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(420, 5);