Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 864 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C22×C5×C7×C9 |
| multiplicative | (ℤ/1349)ˣ, (ℤ/1397)ˣ, (ℤ/1893)ˣ, (ℤ/1917)ˣ, (ℤ/2524)ˣ, (ℤ/2698)ˣ, (ℤ/2794)ˣ, (ℤ/3786)ˣ, (ℤ/3834)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C22×C3×C5×C7 |
| socle, prime radical | C22×C3×C5×C7 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1260 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18)(19,20,21,22,23,24,25);(3,4)1260 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 40);