Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×C9×#339 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C9 | D4 |
| commutator | C4 | C3×C42×C9 |
| Frattini, cosocle | C22×C3×C4 | C22×C32 |
| socle | C22×C32 | C3×C22⋊C4 |
| prime radical | C23×C32 | C3×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C9 ◁ C22×C3×C4×C9 ◁ G | |
550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3)(13,14,15,16)(17,18)(19,21)(20,23)(22,24);(4,5,6,7,8,9,10,11,12)(18,21,23,22)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2081);