Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42×C9 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C3×C9×#405 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8×C9 | C22 |
| commutator | C2 | C2×C3×C42×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C22×C32 | C22×C3×C4 |
| prime radical | C23×C32 | C2×C3×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8×C9 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(4,5,6,7,8,9,10,11,12)(13,14,15,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(19,25)(22,28)(26,31)(29,32)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11036);