Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#10593105 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #64188 |
| commutator, cosocle, hypocenter | C22⋊(Q8⋊C9) | C2×C3 |
| Frattini | C2×C3 | C3×#1020 |
| Fitting | C22×C32×Q8 | D3 |
| socle, prime radical | C23×C32 | S4 |
| LCS | G ▷ C22⋊(Q8⋊C9) | |
| UCS | C1 ◁ C2×C3 | |
271 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
81 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30582);