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 | C33×#532 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C33×C4 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C33 |
| socle | C22×C33 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C33×C4 ◁ G | |
4340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3)(11,13)(14,15)(16,19)(17,18)(20,21);(4,5,6)(15,18)(19,21);(7,8,9)(10,11,12,13);(14,16,17,20)(15,19,18,21)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46050);