Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C33×C4 · Schur multiplier (H2) = C25×C33 |
| direct product | C22×C33×M16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C33×C4 | C22 |
| commutator | C2 | C23×C33×C4 |
| Frattini, cosocle | C4 | C24×C33 |
| socle | C23×C33 | C2×C4 |
| prime radical | C24×C33 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C33×C4 ◁ G | |
4172 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)(13,14,15);(3,4)(16,17,18);(5,6,7,8,9,10,11,12)(19,20,21);(6,10)(8,12)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46084);