Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of #4245 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#4247 |
| Hall | C3×C5×C7 ⋊ D4 |
| Hall | C5×C7 ⋊ D12 |
| Hall | C3×C7 ⋊ C5×D4 |
| Hall | C3×C5 ⋊ C22⋊D7 |
| Sylow | C7 ⋊ C5×D12 |
| Sylow | C3 ⋊ C5×C22⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D3×D7 |
| commutator, cosocle | C2×C3×C7 | C22×C5 |
| Frattini | C2 | C5×D3×D7 |
| Fitting, socle | C2×C3×C5×C7 | C22 |
| hypocenter | C3×C7 | C5×D4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2×C5 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 420 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,4,5)(6,7,9,11,12,10,8)(14,15)(17,19);(7,8)(9,10)(11,12)(13,14,15)(16,17,18,19)135 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#4247, #13132505, #13132506, … |
| maximal normal subgroup of | D5×#4247, #13132505, #13132506, … |
| maximal quotient of | C5×#66562, C5×#66567, C5×#66568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 54);