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×#360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | #102 |
| commutator | C8 | C2×C33×C4 |
| Frattini, cosocle | C2×C8 | C22×C33 |
| prime radical | C33×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C2×C33×C4 ◁ C2×C33×C8 ◁ G | |
812 conjugacy classes of subgroups; only the 420 normal subgroups are shown.
This group has 812 conjugacy classes of subgroups — too many to draw. Only its 420 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(11,15,16,24)(12,18)(13,21)(17,19,22,25)(20,23);(4,5,6)(10,11)(12,19)(13,22)(14,16)(15,23)(17,21)(18,24)(20,25);(7,8,9)432 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12216);