Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C9 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×C3×C9×#99 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C9 | C22⋊C4 |
| commutator | C22 | C22×C3×C4×C9 |
| Frattini, cosocle | C23×C3 | C23×C32 |
| socle | C22×C32 | C3×C22⋊C4 |
| prime radical | C22×C32×D4 | C2×C3 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ C24×C3×C9 ◁ G | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(11,12,13);(3,4)(5,9,10,7)(6,8)(14,15,16,17,18,19,20,21,22);(3,5)(4,7)(6,10)(8,9)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11048);