Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C52 · Schur multiplier (H2) = C2×C3×C5 |
| direct product | C32×C52×D4 |
| unambiguous semidirect product | C4⋊(C2×C32×C52) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32×C52 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C32×C52 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C52 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 900 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3)(7,8,9,10,11)(17,18,19,20);(4,5,6)(12,13,14,15,16)(18,20)1125 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 553);