Finite Group DatabasebetaFinite Group Databasebeta| order | 210 |
| isoclinism | isoclinic class of D105 |
| automorphism | Aut(G) = D3×AGL(1,𝔽5)×AGL(1,𝔽7) (order 5040) · Inn(G) = G · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D105 |
| unambiguous semidirect product | C3⋊D35 |
| unambiguous semidirect product | C5⋊D21 |
| unambiguous semidirect product | C7⋊D15 |
| Fitting, socle, derived, Hall | C3×C5×C7 ⋊ C2 |
| Hall | C5×C7 ⋊ D3 |
| Hall | C3×C7 ⋊ D5 |
| Hall | C3×C5 ⋊ D7 |
| Sylow | C7 ⋊ D15 |
| Sylow | C5 ⋊ D21 |
| Sylow | C3 ⋊ D35 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C5×C7 | C2 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 | |
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:
(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15);(2,3)(5,8)(6,7)(10,15)(11,14)(12,13)Polycyclic presentation:
2,3,5,7|b^a=b^2,c^a=c^4,d^a=d^654 conjugacy classes — view the full character table.
| maximal subgroup of | D7×D15, D5×D21, D3×D35, … |
| maximal normal subgroup of | D7×D15, D5×D21, D3×D35, … |
| maximal quotient of | Dic105, C2×D105, D315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(210, 11);