Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C7×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×C7×#170 |
| derived, Sylow | C5 ⋊ C3×C7×C8 |
| Hall | C3×C5×C7 ⋊ C8 |
| Hall | C5×C7 ⋊ C3×C8 |
| Hall | C3×C5 ⋊ C7×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C7 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C3×C7×C8 |
| Frattini | C2 | C3×C7×AGL(1,𝔽5) |
| Fitting, socle, prime radical | C2×C3×C5×C7 | C4 |
| cosocle | Dic5 | C2×C3×C7 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C3×C7 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 840 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(4,5,6,7,8,9,10)(12,13,14,15)(16,17,18,20,19,21,22,23);(11,12,13,15,14)210 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D7×#170, C3×#10587920, C3×#10587921, … |
| maximal normal subgroup of | C3×D7×#170, C3×#10587920, C3×#10587921, … |
| maximal quotient of | C3×C7×#674, C2×C3×C7×#170 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 38);