Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C23×C3 |
| direct product | C3×C9×#453 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | #120 |
| commutator | C2×C4 | C23×C3×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C2×C32 | C3×#120 |
| prime radical | C32×#453 | C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C2×C3×C4×C9 ◁ G | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3)(14,17)(15,19)(18,20);(4,5,6,7,8,9,10,11,12)(13,14)(15,20)(16,18)(17,19);(15,19)(17,20)432 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11180);