Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | D9×#131⋊C3 |
| Fitting | C9×#131 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | A4×D9 |
| commutator, hypocenter | Q8×C9 | C2×C3×C4 |
| Frattini | C3×C4 | C2×D3×A4 |
| Fitting | C9×#131 | C2×C3 |
| socle | C2×C3 | C4×D3×A4 |
| prime radical | D9×#41⋊C3 | C2 |
| cosocle | C9×#41 | C22×C3 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C8 | |
242 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13201);