Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C5×C7 ⋊ Q8 |
| Hall | C5×C7 ⋊ Dic6 |
| Hall | C3×C7 ⋊ Dic10 |
| Hall | C3×C5 ⋊ Dic14 |
| Sylow | C7 ⋊ #1204 |
| Sylow | C5 ⋊ #4248 |
| Sylow | C3 ⋊ Dic70 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×D35 |
| commutator, Fitting, socle, prime radical, cosocle | C2×C3×C5×C7 | C22 |
| hypocenter | C3×C5×C7 | Q8 |
| LCS | G ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 840 |
Polycyclic presentation:
2^3,3,5,7|a^2=c,b^2=c,b^a=bc,e^a=e^4,f^a=f^6,d^b=d^2111 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#1204, #13132510, D5×#4248, … |
| maximal normal subgroup of | D7×#1204, #13132510, D5×#4248, … |
| maximal quotient of | #13132219, #13132220, Dic3⋊Dic35, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 83);