Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of C7⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×C7⋊#170 |
| derived, Hall | C5×C7 ⋊ C3×C8 |
| Hall | C3×C5×C7 ⋊ C8 |
| Hall | C3×C7 ⋊ #170 |
| Hall | C3×C5 ⋊ C7⋊C8 |
| Sylow | C7 ⋊ C3×#170 |
| Sylow | C5 ⋊ C3×C7⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C7⋊AGL(1,𝔽5) |
| commutator, hypocenter | C5×C7 | C3×C8 |
| Frattini | C2 | C3×C7⋊AGL(1,𝔽5) |
| Fitting, socle, prime radical | C2×C3×C5×C7 | C4 |
| cosocle | C7×Dic5 | C2×C3 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C2×C3 | |
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)(5,6)(7,8)(9,10)(12,13,14,15)(16,17,18,20,19,21,22,23);(4,5,7,9,10,8,6)(11,12,15,13,14)102 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, 39);