Finite Group DatabasebetaFinite Group Databasebeta| order | 5004 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊(C139⋊C9) |
| unambiguous semidirect product | C139⋊(C22⋊C9) |
| derived, Hall | C22×C139 ⋊ C9 |
| Sylow | C139 ⋊ C22⋊C9 |
| Sylow | C22 ⋊ C139⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C22⋊(C139⋊C3) |
| commutator, hypocenter | C22×C139 | C9 |
| Fitting, socle | C22×C3×C139 | C3 |
| LCS | G ▷ C22×C139 | |
| UCS | C1 ◁ C3 | |
16 conjugacy classes of subgroups.
Polycyclic presentation:
3,2^2,139,3|a^3=e,b^a=c,c^a=bc,d^a=d^42564 conjugacy classes — view the full character table.
| maximal subgroup of | C5×C22⋊(C139⋊C9), C22⋊#22576804, C22⋊#22576805, … |
| maximal normal subgroup of | C5×C22⋊(C139⋊C9), C7×C22⋊(C139⋊C9) |
| maximal quotient of | C5×C22⋊(C139⋊C9), C22⋊#22576804, C22⋊#22576805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5004, 12);