Finite Group DatabasebetaFinite Group Databasebeta| order | 5598 |
| automorphism | Aut(G) order 5206140 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D2799 |
| unambiguous semidirect product | C9⋊D311 |
| unambiguous semidirect product | C311⋊D9 |
| Fitting, derived, Hall | C9×C311 ⋊ C2 |
| Sylow | C311 ⋊ D9 |
| Sylow | C9 ⋊ D311 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C9×C311 | C2 |
| Frattini | C3 | D933 |
| socle | C3×C311 | D3 |
| LCS | G ▷ C9×C311 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,311,3|b^3=d^2,b^a=b^2d,c^a=c^310,d^a=d^21401 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D9×D311, C2×D2799, C5×D2799, … |
| maximal normal subgroup of | D9×D311, C2×D2799, C5×D2799, … |
| maximal quotient of | Dic2799, C2×D2799, C5×D2799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5598, 4);