Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C9 · Schur multiplier (H2) = C25×C3 |
| direct product | C3×C9×#547 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C9 | C24 |
| commutator | C22 | C24×C3×C9 |
| Frattini, cosocle | C22×C3 | C24×C32 |
| socle | C22×C32 | C24×C3 |
| prime radical | C32×#547 | C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ G | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3)(14,16)(17,18,21,22)(19,20,23,24);(4,5,6,7,8,9,10,11,12)(18,22)(20,24);(13,14)(15,16)(17,19,21,23)(18,20,22,24);(17,20)(18,23)(19,22)(21,24)675 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20686);