Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C33 · Schur multiplier (H2) = C24×C33 |
| direct product | C33×#529 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C33×C4 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C33 |
| socle | C22×C33 | C24 |
| prime radical | C33×#121 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C33×C4 ◁ G | |
3276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3)(10,11)(12,15)(13,14)(16,17)(18,19)(20,22)(21,23)(24,25);(4,5,6)(11,14)(15,17)(18,20,21,24)(19,22,23,25);(7,8,9)(19,23)(22,25);(10,12,13,16)(11,15,14,17)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46047);