Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C5 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×C5×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4×C5 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C32×C5 |
| socle | C2×C32×C5 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4×C5 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(7,8,9,10,11)(12,13)(14,17)(15,16)(18,19);(4,5,6)(13,16)(17,19);(12,14,15,18)(13,17,16,19)450 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 762);