Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C5×C7 ⋊ D4 |
| Hall | C5×C7 ⋊ D12 |
| Hall | C3×C7 ⋊ D20 |
| Hall | C3×C5 ⋊ C22⋊D7 |
| Sylow | C7 ⋊ D60 |
| Sylow | C5 ⋊ #4247 |
| Sylow | C3 ⋊ #63307 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×D15 |
| commutator, Fitting, socle, cosocle | C2×C3×C5×C7 | C22 |
| hypocenter | C3×C5×C7 | D4 |
| LCS | G ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 420 |
Polycyclic presentation:
2^3,3,5,7|a^2=c,b^a=bc,f^a=f^6,d^b=d^2,e^b=e^487 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63307, #13132503, D5×#4247, … |
| maximal normal subgroup of | D3×#63307, #13132503, D5×#4247, … |
| maximal quotient of | #13132130, #13132135, #13132136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 61);