Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42×C9 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C3×C9×H(3,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C9 | C42 |
| commutator | C4 | C3×C42×C9 |
| Frattini, cosocle | C22×C3×C4 | C22×C32 |
| socle | C2×C32 | C3×#95 |
| prime radical | C23×C32 | C3×D4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C9 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3)(13,14)(15,21,23,27)(16,19)(17,20)(18,22,25,28)(24,26);(4,5,6,7,8,9,10,11,12)(13,15)(14,18,20,25)(16,22)(17,23)(19,21,26,27)(24,28)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2075);