Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 6912 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2×C3 |
| abelian | C2×C3×C9×C32 |
| multiplicative | (ℤ/1843)ˣ, (ℤ/2619)ˣ, (ℤ/3686)ˣ, (ℤ/5238)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3×C16 | C22×C32 |
| socle, prime radical | C22×C32 | C3×C16 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 46 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 2 generators on 46 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(35,36,37,38,39,40,41,42,43);(33,34)(44,45,46)1728 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2171);