Finite Group DatabasebetaFinite Group Databasebeta| order | 5580 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊(ΓL(1,𝔽32)⋊C9) |
| unambiguous semidirect product | ΓL(1,𝔽32)⋊(C22⋊C9) |
| unambiguous semidirect product | (C22⋊(C31⋊C9))⋊C5 |
| derived, Hall | C22×C31 ⋊ C5×C9 |
| Hall | C22⋊(C31⋊C9) ⋊ C5 |
| Hall | C22×ΓL(1,𝔽32) ⋊ C9 |
| Hall | ΓL(1,𝔽32) ⋊ C22⋊C9 |
| Sylow | C31 ⋊ C5×C22⋊C9 |
| Sylow | C22 ⋊ ΓL(1,𝔽32)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C22⋊((C31⋊C3)⋊C5) |
| commutator, hypocenter | C22×C31 | C5×C9 |
| Fitting, socle | C22×C3×C31 | C3×C5 |
| LCS | G ▷ C22×C31 | |
| UCS | C1 ◁ C3 | |
32 conjugacy classes of subgroups.
Polycyclic presentation:
3,5,2^2,31,3|a^3=f,c^a=d,d^a=cd,e^a=e^5,e^b=e^284 conjugacy classes — view the full character table.
| maximal subgroup of | C5×C22⋊(ΓL(1,𝔽32)⋊C9) |
| maximal normal subgroup of | C5×C22⋊(ΓL(1,𝔽32)⋊C9) |
| maximal quotient of | C22⋊((C31⋊C9)⋊C25), C5×C22⋊(ΓL(1,𝔽32)⋊C9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5580, 24);