Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C2×C33 |
| direct product | C33×#362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | #102 |
| commutator | C8 | C2×C33×C4 |
| Frattini, cosocle | C2×C8 | C22×C33 |
| prime radical | C22×C33 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C2×C33×C4 ◁ C2×C33×C8 ◁ G | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3)(10,11,14,18,15,20,30,19)(12,21,23,34,24,38,29,37)(13,25,27,17,28,35,41,36)(16,26,31,22,33,39,32,40);(4,5,6)(10,12,15,24)(11,16,20,33)(13,26,28,39)(14,29,30,23)(17,34,36,37)(18,32,19,31)(21,25,38,35)(22,27,40,41);(7,8,9)432 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12218);