Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C33 · Schur multiplier (H2) = C25×C33 |
| direct product | C2×C33×C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C33 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C33 |
| prime radical | C24×C33 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C33 ◁ G | |
4508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(15,16,17);(4,6)(7,8,10,12)(9,13,14,11)(18,19,20);(7,9,10,14)(8,11,12,13)(21,22,23);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46041);