Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C9 · Schur multiplier (H2) = C23×C3 |
| direct product | C3×C9×#408 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C9 | C23 |
| commutator | C22 | C22×C3×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C22×C32 | C22×C3×C4 |
| prime radical | C2×C32×#41 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C9 ◁ G | |
790 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)(13,14)(15,16)(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)(14,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(19,25)(22,28)(26,31)(29,32)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11045);