Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4×C42⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C2×C33×C4 |
| direct product | C3×A4×C42⋊C3 |
| unambiguous semidirect product | (C22×C42⋊C3)⋊C32 |
| unambiguous semidirect product | (C42×A4)⋊C32 |
| unambiguous semidirect product | (C22⋊(C42⋊C3))⋊C32 |
| Fitting | C22×C3×C42 ⋊ C32 |
| derived, Sylow | C22×C42 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4×C42⋊C3 |
| commutator, cosocle, hypocenter | C22×C42 | C33 |
| Frattini | C22 | C3×A42 |
| Fitting | C22×C3×C42 | C32 |
| socle | C24×C3 | C3×A4 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 ◁ C3 | |
302 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32235);