Finite Group DatabasebetaFinite Group Databasebeta| order | 210 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C3×C4×D3 (order 144) · Inn(G) = D3 · Out(G) = C2×C3×C4 |
| holomorph | G ⋊ Aut(G) = D32×AGL(1,𝔽5)×AGL(1,𝔽7) |
| homology | Abelianization (H1) = C2×C5×C7 · Schur multiplier (H2) = C1 |
| direct product | C5×D3×C7 |
| unambiguous semidirect product | C3⋊(C2×C5×C7) |
| Fitting, socle, Hall | C3×C5×C7 ⋊ C2 |
| derived, Sylow | C3 ⋊ C2×C5×C7 |
| Hall | C3×C7 ⋊ C2×C5 |
| Hall | C3×C5 ⋊ C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C7 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C5×C7 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C7 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C5×C7 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 105 |
A minimal faithful permutation representation — 2 generators on 15 points:
(2,3)(4,5,6,7,8)(9,10,11,12,13,14,15);(1,2,3)105 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D3×D7, D3×C7×D5, D3×D35, … |
| maximal normal subgroup of | C5×D3×D7, D3×C7×D5, D3×D35, … |
| maximal quotient of | C5×C7×Dic3, C2×C5×D3×C7, C5×C7×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(210, 8);