Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | A4×C3⋊#226 |
| unambiguous semidirect product | C3⋊(A4×#226) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×C3⋊S4 |
| commutator, cosocle, hypocenter | C22×C3×SL(2,𝔽3) | C2×C3 |
| Fitting | C22×C3×Q8 | C3×D3 |
| socle | C23×C3 | C3×S4 |
| prime radical | C3×A4×SL(2,𝔽3) | C2 |
| LCS | G ▷ C22×C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
306 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46384);