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) = C25×C33 |
| direct product | C33×#386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C33 | C23 |
| commutator | C22 | C22×C33×C4 |
| Frattini, cosocle | C23 | C23×C33 |
| prime radical | C22×C33×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C33 ◁ G | |
4004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3)(11,13)(14,15,16,17)(19,21);(4,5,6)(14,15)(16,17)(18,19,20,21);(7,8,9)(10,11)(12,13)(14,15)(16,17)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 29963);