Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C9 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C3×C9×#353 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | #99 |
| commutator | C2×C4 | C2×C3×C4×C9 |
| Frattini, cosocle | C2×C3×D4 | C22×C32 |
| socle | C2×C32 | C3×#99 |
| prime radical | C32×#127 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C22×C3×C9 ◁ C2×C3×D4×C9 ◁ G | |
430 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3)(13,14)(15,18,20,17)(16,19);(4,5,6,7,8,9,10,11,12)(14,17)(15,20)(18,19)351 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2123);