Finite Group DatabasebetaFinite Group Databasebeta| order | 210 |
| isoclinism | isoclinic class of D15 |
| automorphism | Aut(G) = C2×C3×D3×AGL(1,𝔽5) (order 720) · Inn(G) = D15 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×D15 |
| unambiguous semidirect product | C3⋊(C7×D5) |
| unambiguous semidirect product | C5⋊(D3×C7) |
| Fitting, socle, Hall | C3×C5×C7 ⋊ C2 |
| derived, Hall | C3×C5 ⋊ C2×C7 |
| Hall | C5×C7 ⋊ D3 |
| Hall | C3×C7 ⋊ D5 |
| Sylow | C5 ⋊ D3×C7 |
| Sylow | C3 ⋊ C7×D5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | D15 |
| commutator, cosocle, hypocenter | C3×C5 | C2×C7 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C7 | C2 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ 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)(5,6)(7,8)(9,10,11,12,13,14,15);(1,2,3)(4,5,7,8,6)63 conjugacy classes — view the full character table.
| maximal subgroup of | D7×D15, D3×C7×D5, D15⋊D7, … |
| maximal normal subgroup of | D7×D15, D3×C7×D5, D15⋊D7, … |
| maximal quotient of | C7×Dic15, C2×C7×D15, C7×D45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(210, 10);