Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 384 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C5×C8×C9 |
| multiplicative | (ℤ/779)ˣ, (ℤ/803)ˣ, (ℤ/1107)ˣ, (ℤ/1558)ˣ, (ℤ/1606)ˣ, (ℤ/2214)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3×C4 | C22×C3×C5 |
| socle, prime radical | C22×C3×C5 | C3×C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,4,5,6,7,8)(11,12,13,14,15,16,17,18,19)(20,21,22,23,24);(9,10)720 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 84);