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) = C22×C3×C4 |
| direct product | C3×C9×#382 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C9 | C23 |
| commutator | C22 | C22×C3×C4×C9 |
| Frattini, cosocle | C23×C3 | C23×C32 |
| socle, prime radical | C23×C32 | C23×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C9 ◁ G | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3)(14,15)(17,18,19,20)(23,27)(25,28);(4,5,6,7,8,9,10,11,12)(13,14,16,15)(21,22,24,26)(23,25,27,28);(13,15,16,14)(21,23)(22,25)(24,27)(26,28)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10967);