Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C9 · Schur multiplier (H2) = C23×C3 |
| direct product | C3×C9×#421 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C9 | C2×D4 |
| commutator | C4 | C22×C3×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C2×C32 | C2×C3×C22⋊C4 |
| prime radical | C2×C32×#41 | C2×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C9 ◁ C22×C3×C4×C9 ◁ G | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3)(13,14,17,20,18,21,16,25)(15,22,24,19,23,28,26,27);(4,5,6,7,8,9,10,11,12)(13,15)(14,19)(16,26)(17,24)(18,23)(20,28)(21,27)(22,25);(15,23)(19,27)(22,28)(24,26)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11084);