Finite Group DatabasebetaFinite Group Databasebeta| order | 5642 |
| automorphism | Aut(G) order 6093360 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D2821 |
| unambiguous semidirect product | C7⋊D403 |
| unambiguous semidirect product | C13⋊D217 |
| unambiguous semidirect product | C31⋊D91 |
| Fitting, socle, derived, Hall | C7×C13×C31 ⋊ C2 |
| Hall | C13×C31 ⋊ D7 |
| Hall | C7×C31 ⋊ D13 |
| Hall | C7×C13 ⋊ D31 |
| Sylow | C31 ⋊ D91 |
| Sylow | C13 ⋊ D217 |
| Sylow | C7 ⋊ D403 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C13×C31 | C2 |
| LCS | G ▷ C7×C13×C31 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
Polycyclic presentation:
2,7,13,31|b^a=b^6,c^a=c^12,d^a=d^301412 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D31×D91, D13×D217, D7×D403, … |
| maximal normal subgroup of | D31×D91, D13×D217, D7×D403, … |
| maximal quotient of | Dic2821, C2×D2821, C3×D2821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5642, 8);