Finite Group DatabasebetaFinite Group Databasebeta| order | 140 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×C4×AGL(1,𝔽7) (order 336) · Inn(G) = D7 · Out(G) = C2×C3×C4 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽5)×#72210376 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×C5×D7 |
| unambiguous semidirect product | C7⋊(C22×C5) |
| unambiguous semidirect product | (C2×C7)⋊(C2×C5) |
| Fitting, socle | C2×C5×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C5 |
| Hall | C5×C7 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D7 |
| commutator, cosocle, hypocenter | C7 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C5×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C5 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 70 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2)(3,4,5,6,7)(9,10)(11,12)(13,14);(1,2)(8,9,11,13,14,12,10)50 conjugacy classes — view the full character table.
| maximal subgroup of | D7×Dic5, #63305, #63306, … |
| maximal normal subgroup of | D7×Dic5, #63305, #63306, … |
| maximal quotient of | C5×Dic14, C4×C5×D7, C5×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(140, 8);