Finite Group DatabasebetaFinite Group Databasebeta| order | 5628 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C67⋊C3)⋊Dic7 |
| derived, Hall | C7×C67 ⋊ C3×C4 |
| Hall | Dic469 ⋊ C3 |
| Hall | C7×C67⋊C3 ⋊ C4 |
| Hall | C67⋊C3 ⋊ Dic7 |
| Sylow | C67 ⋊ C3×Dic7 |
| Sylow | C7 ⋊ (C67⋊C3)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | (C67⋊C3)⋊D7 |
| commutator, hypocenter | C7×C67 | C3×C4 |
| Fitting, socle | C2×C7×C67 | C2×C3 |
| LCS | G ▷ C7×C67 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,7,67,2|a^2=e,c^a=c^6,d^a=d^66,d^b=d^37184 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5628, 21);