Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 2304 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C4 |
| abelian | C42×C5×C9 |
| multiplicative | (ℤ/905)ˣ, (ℤ/925)ˣ, (ℤ/1810)ˣ, (ℤ/1850)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C22×C3 | C22×C3×C5 |
| socle, prime radical | C22×C3×C5 | C22×C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4)(9,10,11,12,13,14,15,16,17)(18,19,20,21,22);(5,6,7,8)720 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 81);