Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8×C9 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C3×C9×C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4×C9 | C22 |
| commutator | C2 | C22×C3×C8×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle, prime radical | C23×C32 | C2×C3×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4×C9 ◁ G | |
690 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);(4,6)(7,8,9,11,10,12,13,14)(18,19,20,21,22,23,24,25,26);(3,4,5,6)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11087);