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) = C22 |
| direct product | C5×C9×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5×C9 | C22 |
| commutator | C2 | C23×C5×C9 |
| Frattini, cosocle | C2×C3 | C23×C3×C5 |
| socle | C2×C3×C5 | C23×C3 |
| prime radical | C3×C5×#41 | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5×C9 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3,4,5)(6,7,8,9,10,11,12,13,14)(15,16)(17,20)(18,19)(21,22);(16,19)(20,22);(15,17,18,21)(16,20,19,22)450 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 188);