Finite Group DatabasebetaFinite Group Databasebeta| order | 5550 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C3⋊(C5⋊D185) |
| unambiguous semidirect product | C5⋊D555 |
| unambiguous semidirect product | C37⋊(C3⋊(C5⋊D5)) |
| Fitting, socle, derived, Hall | C3×C52×C37 ⋊ C2 |
| Hall | C52×C37 ⋊ D3 |
| Hall | C3×C37 ⋊ C5⋊D5 |
| Hall | C3×C52 ⋊ D37 |
| Sylow | C37 ⋊ C3⋊(C5⋊D5) |
| Sylow | C52 ⋊ D111 |
| Sylow | C3 ⋊ C5⋊D185 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C52×C37 | C2 |
| LCS | G ▷ C3×C52×C37 | |
| UCS | C1 | |
64 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,5^2,37|b^a=b^2,c^a=c^4,d^a=d^4,e^a=e^361389 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D37×C3⋊(C5⋊D5), C5⋊D5×D111, D3×C5⋊D185, … |
| maximal normal subgroup of | D37×C3⋊(C5⋊D5), C5⋊D5×D111, D3×C5⋊D185, … |
| maximal quotient of | C3⋊#16814867, C2×C3⋊(C5⋊D185), C5⋊D1665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5550, 40);