Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C9×#367 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C9 | D8 |
| commutator | C8 | C2×C3×C4×C9 |
| Frattini, cosocle | C2×C3×C8 | C22×C32 |
| socle, prime radical | C22×C32 | C3×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ C2×C3×C4×C9 ◁ C2×C3×C8×C9 ◁ G | |
230 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3)(13,14,15,16)(18,22)(19,26)(20,27)(23,28)(24,32)(25,31)(29,30);(4,5,6,7,8,9,10,11,12)(17,18,23,32,20,24,19,22,21,25,26,30,27,29,28,31)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2165);