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) = C22×C3 |
| direct product | C3×C9×#504 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C16 | C22 |
| commutator | C2 | C22×C3×C8×C9 |
| Frattini, cosocle | C3×C8 | C23×C32 |
| socle | C2×C32 | C22×C3×C8 |
| prime radical | C32×#41 | C3×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C16 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,3)(13,14,16,20,17,21,26,32,18,22,27,33,28,34,38,42)(15,19,24,30,25,31,37,41,23,29,35,39,36,40,43,44);(4,5,6,7,8,9,10,11,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,30)(21,31)(22,29)(26,37)(27,35)(28,36)(32,41)(33,39)(34,40)(38,43)(42,44);(15,23)(19,29)(24,35)(25,36)(30,39)(31,40)(37,43)(41,44)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11333);