Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C9 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×D4×C9 |
| unambiguous semidirect product | C4⋊(C22×C5×C9) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C9 | C22 |
| commutator | C2 | C23×C5×C9 |
| Frattini, cosocle | C2×C3 | C23×C3×C5 |
| socle | C22×C3×C5 | C22×C3 |
| prime radical | C2×C3×C5×D4 | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C9 ◁ G | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(7,8,9,10,11)(12,13,14,15,16,17,18,19,20);(3,4,5,6);(4,6)450 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 186);