Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×C4×C5×3−1+2 |
| unambiguous semidirect product | (C2×C4×C9)⋊(C3×C5) |
| unambiguous semidirect product | (C2×C4×C5×C9)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C5 | C32 |
| commutator | C3 | C2×C32×C4×C5 |
| Frattini, cosocle | C2×C3 | C22×C32×C5 |
| socle | C22×C3×C5 | C2×C32 |
| prime radical | C22×C32×C5 | C2×C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C5 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(7,8,9,10,11)(12,13,15,14,17,19,18,16,20);(3,4,5,6)(13,16,17)(15,19,20)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 245);