Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C9 · Schur multiplier (H2) = C27×C3 |
| direct product | C2×C3×C9×#120 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C9 | C23 |
| commutator | C22 | C24×C3×C9 |
| Frattini, cosocle | C22×C3 | C24×C32 |
| socle | C23×C32 | C23×C3 |
| prime radical | C2×C32×#120 | C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C9 ◁ G | |
3310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2)(11,12,13);(3,4)(5,7)(6,8)(9,10)(14,15,16,17,18,19,20,21,22);(4,7)(8,10);(4,8)(7,10)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20608);