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) = C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×Dic7 |
| unambiguous semidirect product | C7⋊(C4×C5) |
| derived, Sylow | C7 ⋊ C4×C5 |
| Hall | C5×C7 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D7 |
| commutator, hypocenter | C7 | C4×C5 |
| Frattini | C2 | C5×D7 |
| Fitting, socle, prime radical | C2×C5×C7 | C2 |
| cosocle | C2×C7 | C2×C5 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C5 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 140 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,12,13,14,15,16);(6,8,7,9)(11,16)(12,15)(13,14)50 conjugacy classes — view the full character table.
| maximal subgroup of | D5×Dic7, #63304, #63307, … |
| maximal normal subgroup of | D5×Dic7, #63304, #63307, … |
| maximal quotient of | C5×C7⋊C8, C2×C5×Dic7, C3×C5×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(140, 2);