Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C9 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C3×C9×#106 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C9 | D4 |
| commutator | C4 | C22×C3×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle, prime radical | C23×C32 | C3×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C9 ◁ C22×C3×C4×C9 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2)(15,16,17);(3,4,5,6)(8,11)(9,13)(12,14)(18,19,20,21,22,23,24,25,26);(7,8,9,11,10,12,13,14)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11096);