Finite Group DatabasebetaFinite Group Databasebeta| order | 1120 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 516096 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C26 |
| abelian | C23×C4×C5×C7 |
| multiplicative | (ℤ/2552)ˣ, (ℤ/2840)ˣ, (ℤ/3408)ˣ, (ℤ/3828)ˣ, (ℤ/4260)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C24×C5×C7 |
| socle, prime radical | C24×C5×C7 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 1120 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3,4)(11,12,13,14,15)(16,17,18,19,20,21,22);(5,6);(7,8);(9,10)1120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1120, 1072);