Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C3×C5×C7 ⋊ C2 |
| Hall | C3×C5×C7 ⋊ D4 |
| Hall | C5×C7 ⋊ #67 |
| Hall | C3×C7 ⋊ C22⋊D5 |
| Hall | C3×C5 ⋊ C22⋊D7 |
| Sylow | C7 ⋊ #1220 |
| Sylow | C5 ⋊ #4268 |
| Sylow | C3 ⋊ C22⋊D35 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D105 |
| commutator, socle, cosocle | C2×C3×C5×C7 | C22 |
| Fitting | C22×C3×C5×C7 | C2 |
| hypercenter | C22 | D105 |
| hypocenter | C3×C5×C7 | D4 |
| LCS | G ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 420 |
Polycyclic presentation:
2^3,3,5,7|b^a=bc,d^a=d^2,e^a=e^4,f^a=f^6213 conjugacy classes — view the full character table.
| maximal subgroup of | #13132700, #13132703, D7×#1220, … |
| maximal normal subgroup of | #13132700, #13132703, D7×#1220, … |
| maximal quotient of | #13132361, #13132363, #13132364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 130);