Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3×C9 · Schur multiplier (H2) = C210×C3 |
| direct product | C23×C3×D4×C9 |
| unambiguous semidirect product | C4⋊(C24×C3×C9) |
| subgroup | G/H | |
|---|---|---|
| center | C24×C3×C9 | C22 |
| commutator | C2 | C25×C3×C9 |
| Frattini, cosocle | C2×C3 | C25×C32 |
| socle | C24×C32 | C22×C3 |
| prime radical | C23×C32×D4 | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3×C9 ◁ G | |
6810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2)(11,12,13);(3,4)(14,15,16,17,18,19,20,21,22);(5,6);(7,8,9,10);(8,10)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31077);