Finite Group DatabasebetaFinite Group Databasebeta| order | 140 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽5) (order 240) · Inn(G) = D5 · Out(G) = C23×C3 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽7)×#11717375 |
| homology | Abelianization (H1) = C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×Dic5 |
| derived, Sylow | C5 ⋊ C4×C7 |
| Hall | C5×C7 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D5 |
| commutator, hypocenter | C5 | C4×C7 |
| Frattini | C2 | C7×D5 |
| Fitting, socle, prime radical | C2×C5×C7 | C2 |
| cosocle | C2×C5 | C2×C7 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C7 | |
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);(8,10,9,11)(13,16)(14,15)56 conjugacy classes — view the full character table.
| maximal subgroup of | C7×#170, C7⋊#170, D7×Dic5, … |
| maximal normal subgroup of | C7×#170, C7⋊#170, D7×Dic5, … |
| maximal quotient of | C7×#168, C2×C7×Dic5, C3×C7×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(140, 1);