Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C52×C9 · Schur multiplier (H2) = C5 |
| direct product | C52×Q8×C9 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C52×C9 | C22 |
| commutator | C2 | C22×C52×C9 |
| Frattini, cosocle | C2×C3 | C22×C3×C52 |
| socle, prime radical | C2×C3×C52 | C22×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C52×C9 ◁ G | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1800 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5)(11,12,13,14)(15,18,17,16)(19,20,21,22,23,24,25,26,27);(6,7,8,9,10)(11,15,13,17)(12,16,14,18)1125 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 214);