Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of D15⋊D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D21⋊Dic5 |
| derived, Hall | C3×C5×C7 ⋊ C2×C4 |
| Hall | C5×C7 ⋊ C4×D3 |
| Hall | C3×C7 ⋊ C2×Dic5 |
| Hall | C3×C5 ⋊ C4×D7 |
| Sylow | C7 ⋊ D3×Dic5 |
| Sylow | C5 ⋊ D21⋊C4 |
| Sylow | C3 ⋊ D7×Dic5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D15⋊D7 |
| commutator, hypocenter | C3×C5×C7 | C2×C4 |
| Fitting, socle, cosocle | C2×C3×C5×C7 | C22 |
| prime radical | C2×C5×D21 | C2 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 420 |
Polycyclic presentation:
2^3,3,5,7|b^2=c,d^a=d^2,f^a=f^6,d^b=d^2,e^b=e^478 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×Dic5, D5×D21⋊C4, #13132491, … |
| maximal normal subgroup of | D3×D7×Dic5, D5×D21⋊C4, #13132491, … |
| maximal quotient of | D21⋊#168, #13132227, #13132240, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 84);