Finite Group DatabasebetaFinite Group Databasebeta| order | 140 |
| isoclinism | isoclinic class of C7⋊AGL(1,𝔽5) |
| automorphism | Aut(G) = AGL(1,𝔽5)×AGL(1,𝔽7) (order 840) · Inn(G) = G · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊AGL(1,𝔽5) |
| Fitting, socle, derived, Hall | C5×C7 ⋊ C4 |
| Sylow | C7 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C7 | C4 |
| prime radical, cosocle | C7×D5 | C2 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 35 |
A minimal faithful permutation representation — 2 generators on 12 points:
(2,3)(4,5)(6,7)(9,10,11,12);(1,2,4,6,7,5,3)(8,9,12,10,11)Polycyclic presentation:
2^2,5,7|a^2=b,c^a=c^3,d^a=d^6,c^b=c^417 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(140, 6);