Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #63306⋊C3 |
| Hall | #63306 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ D4 |
| Hall | C5×C7 ⋊ C3×D4 |
| Hall | ΓL(1,𝔽8) ⋊ C22⋊D5 |
| Sylow | C7 ⋊ C3×C22⋊D5 |
| Sylow | C5 ⋊ C4⋊AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×AGL(1,𝔽7) |
| commutator, Fitting, socle, cosocle | C2×C5×C7 | C22×C3 |
| hypocenter | C5×C7 | C3×D4 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 140 |
Polycyclic presentation:
2^2,3,2,5,7|b^2=d,b^a=bd,f^a=f^6,e^b=e^4,f^c=f^447 conjugacy classes — view the full character table.
| maximal subgroup of | #10587939⋊C3, #10587944⋊C3, #10587946⋊C3, … |
| maximal normal subgroup of | #10587939⋊C3, #10587944⋊C3, #10587946⋊C3, … |
| maximal quotient of | #10587842⋊C3, #10587846⋊C3, #10587847⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 20);