Finite Group DatabasebetaFinite Group Databasebeta| order | 210 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×C4×AGL(1,𝔽7) (order 336) · Inn(G) = D7 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×D7 |
| Fitting, socle, Hall | C3×C5×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C2×C3×C5 |
| Hall | C5×C7 ⋊ C2×C3 |
| Hall | C3×C7 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | D7 |
| commutator, cosocle, hypocenter | C7 | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C5 | |
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)(10,11)(12,13)(14,15);(9,10,12,14,15,13,11)75 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5×D7, C5×D3×D7, D7×D15, … |
| maximal normal subgroup of | C3×D5×D7, C5×D3×D7, D7×D15, … |
| maximal quotient of | C3×C5×Dic7, C2×C3×C5×D7, C5×C9×D7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(210, 5);