Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×C9×#368 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C9 | D8 |
| commutator | C8 | C2×C3×C4×C9 |
| Frattini, cosocle | C2×C3×C8 | C22×C32 |
| socle | C2×C32 | C3×#107 |
| prime radical | C22×C32 | C3×Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C9 ◁ C2×C3×C4×C9 ◁ C2×C3×C8×C9 ◁ G | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,3)(13,14,17,22,19,23,18,33)(15,24,27,39,29,40,28,43)(16,30,31,37,32,44,25,21)(20,34,35,42,36,41,38,26);(4,5,6,7,8,9,10,11,12)(13,15,25,41,18,28,32,42,19,29,31,34,17,27,16,26)(14,20,30,39,22,35,37,40,23,36,44,43,33,38,21,24)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2168);