Finite Group DatabasebetaFinite Group Databasebeta| order | 6660 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C37 · Schur multiplier (H2) = C1 |
| direct product | C3×C37×ΓL(1,𝔽16) |
| unambiguous semidirect product | C3⋊(C3×AGL(1,𝔽5)×C37) |
| Fitting, socle, Hall | C32×C5×C37 ⋊ C4 |
| derived | C3×C5 ⋊ C3×C4×C37 |
| Hall | C32×C37 ⋊ AGL(1,𝔽5) |
| Hall | C5×C37 ⋊ C3×Dic3 |
| Hall | C32×C5 ⋊ C4×C37 |
| Sylow | C32 ⋊ AGL(1,𝔽5)×C37 |
| Sylow | C5 ⋊ C3×Dic3×C37 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C37 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C3×C4×C37 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C37 | C4 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C3×C37 | |
56 conjugacy classes of subgroups.
999 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C37×C3⋊#1035, C3×C5×C37×ΓL(1,𝔽16), C3×C37×C3⋊#1042, … |
| maximal normal subgroup of | C3×C5×C37×ΓL(1,𝔽16) |
| maximal quotient of | C3×C37×C3⋊#1035, C3×C5×C37×ΓL(1,𝔽16), C3×C37×C3⋊#1042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6660, 126);