Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of S42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×S42 |
| unambiguous semidirect product | (A4×S4)⋊(C2×C3) |
| Fitting, socle | C24×C3 ⋊ D32 |
| derived | A42 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | S42 |
| commutator, cosocle, hypocenter | A42 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3 | D32 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C3 | |
601 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 601 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
75 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47851);