Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×D9⋊(#41⋊C3) |
| unambiguous semidirect product | D9⋊(C2×#41⋊C3) |
| Fitting | C2×C9×#41 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D9⋊A4 |
| commutator, cosocle, hypocenter | Q8×C9 | C23×C3 |
| Frattini | C2×C3 | C22×D3×A4 |
| Fitting | C2×C9×#41 | C2×C3 |
| socle | C22×C3 | C2×D3×A4 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C2×C4 | |
673 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 673 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30649);