Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | A4×#3699 |
| Fitting | C22×D4×C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×A4×D9 |
| commutator, cosocle | C23×C9 | C23×C3 |
| Frattini | C2×C3 | C22×D3×A4 |
| Fitting | C22×D4×C9 | C2×C3 |
| socle | C23×C3 | C22×C3×D3 |
| hypercenter | D4 | A4×D9 |
| hypocenter | C22×C9 | C3×#41 |
| LCS | G ▷ C23×C9 ▷ C22×C9 | |
| UCS | C1 ◁ C2 ◁ D4 | |
677 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 677 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30634);