Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C3×C9×D16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C9 | D8 |
| commutator | C8 | C23×C3×C9 |
| Frattini, cosocle | C3×C8 | C23×C32 |
| socle | C22×C32 | C3×D8 |
| prime radical | C2×C32×D16 | C3 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ C2×C3×C4×C9 ◁ C2×C3×C8×C9 ◁ G | |
490 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 250 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2)(19,20,21);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(22,23,24,25,26,27,28,29,30);(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11336);