Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of D3×D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D21⋊C16 |
| derived, Hall | C3×C7 ⋊ C2×C16 |
| Sylow | C7 ⋊ D3×C16 |
| Sylow | C3 ⋊ D7×C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D3×D7 |
| commutator, hypocenter | C3×C7 | C2×C16 |
| Fitting, cosocle | C3×C7×C8 | C22 |
| socle | C2×C3×C7 | C2×C8 |
| prime radical | C2×D21 | C8 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C8 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 336 |
Polycyclic presentation:
2^5,3,7|a^2=c,b^2=c,c^2=d,d^2=e,g^a=g^6,f^b=f^2120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×C16, #13052188, #13052190, … |
| maximal normal subgroup of | D3×D7×C16, #13052188, #13052190, … |
| maximal quotient of | D21⋊C32, #13050845, #13050900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 50);