Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C16 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C3⋊(A4⋊C16) |
| unambiguous semidirect product | C3⋊(C3×A4⋊C16) |
| unambiguous semidirect product | A4⋊(C3×C3⋊C16) |
| derived | C3×A4 ⋊ C3×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | C3⋊S4 |
| commutator, hypocenter | C3×A4 | C3×C16 |
| Frattini | C8 | C3×C3⋊S4 |
| Fitting | C22×C32×C8 | D3 |
| socle | C23×C32 | C3⋊C8 |
| prime radical | C2×C32×A4 | C8 |
| cosocle | C3×C8×A4 | C2×C3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C3×C8 | |
246 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 23076);