Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33×C4 · Schur multiplier (H2) = C23×C33 |
| direct product | C33×#383 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C33 | C23 |
| commutator | C22 | C22×C33×C4 |
| Frattini, cosocle | C23 | C23×C33 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C33 ◁ G | |
1932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3)(10,11,12,13)(16,20)(18,21)(23,26)(24,28);(4,5,6)(14,15,17,19)(16,18,20,21)(22,23)(24,27)(25,26)(28,29);(7,8,9)(14,16)(15,18)(17,20)(19,21)(22,24,25,28)(23,27,26,29)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 29960);