Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| isoclinism | isoclinic class of C5⋊(D9⋊A4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C5⋊(D9⋊A4) |
| unambiguous semidirect product | D45⋊A4 |
| unambiguous semidirect product | (C9⋊A4)⋊D5 |
| Fitting, derived | C22×C5×C9 ⋊ C2×C3 |
| Sylow | C5 ⋊ D9⋊A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C22×C5×C9 | C2×C3 |
| Frattini | C3 | A4×D15 |
| socle | C22×C3×C5 | C3×D3 |
| LCS | G ▷ C22×C5×C9 | |
| UCS | C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2,3,2^2,3,5,3|e^3=g^2,e^a=e^2g,f^a=f^4,g^a=g^2,c^b=d,d^b=cd,e^b=eg56 conjugacy classes — view the full character table.
| maximal subgroup of | A4×C5⋊AGL(1,ℤ/9), D5×D9⋊A4 |
| maximal normal subgroup of | A4×C5⋊AGL(1,ℤ/9), D5×D9⋊A4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 312);