Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C9 · Schur multiplier (H2) = C22×C3×C4 |
| direct product | C3×C9×#410 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C9 | C22⋊C4 |
| commutator | C22 | C22×C3×C4×C9 |
| Frattini, cosocle | C23×C3 | C23×C32 |
| socle | C2×C32 | C2×C3×C22⋊C4 |
| prime radical | C2×C32×#41 | C2×C3 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C9 ◁ C22×C3×C4×C9 ◁ G | |
790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3)(13,14)(15,21,23,18)(16,19)(17,20)(22,27,28,25)(24,26);(4,5,6,7,8,9,10,11,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11051);