Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C33 |
| direct product | C33×#330 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C33 | C22⋊C4 |
| commutator | C2×C4 | C2×C33×C4 |
| Frattini, cosocle | C42 | C22×C33 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C33 ◁ C33×C42 ◁ G | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(11,14,15,16)(18,19,22,24,23,20,21,25);(4,5,6)(10,11)(12,16)(13,15)(14,17)(18,20,23,19)(21,24,22,25);(7,8,9)513 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12186);