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 | C2×C33×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C33×C8 | C22 |
| commutator | C2 | C23×C33×C4 |
| Frattini, cosocle | C4 | C24×C33 |
| socle | C22×C33 | C22×C4 |
| prime radical | C2×C33×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33×C8 ◁ G | |
3724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(19,20,21);(3,4,6,9,7,10,13,16)(5,8,12,15,11,14,17,18)(22,23,24);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18)(25,26,27);(5,11)(8,14)(12,17)(15,18)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46085);