Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 72576 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3 |
| abelian | C23×C3×C5×C9 |
| multiplicative | (ℤ/1463)ˣ, (ℤ/1767)ˣ, (ℤ/1881)ˣ, (ℤ/2079)ˣ, (ℤ/2356)ˣ, (ℤ/2926)ˣ, (ℤ/3348)ˣ, (ℤ/3534)ˣ, (ℤ/3762)ˣ, (ℤ/4158)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C23×C32×C5 |
| socle, prime radical | C23×C32×C5 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 1080 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(7,8,9,10,11,12,13,14,15)(19,20,21,22,23);(3,4)(16,17,18);(5,6)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 360);