Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of S4×C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C3 |
| direct product | S4×C3⋊S4 |
| unambiguous semidirect product | (C3×S4)⋊S4 |
| unambiguous semidirect product | (A4×S4)⋊D3 |
| Fitting, socle | C24×C3 ⋊ D32 |
| derived | C3×A42 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×A42 | C22 |
| Fitting, socle | C24×C3 | D32 |
| LCS | G ▷ C3×A42 | |
| UCS | C1 | |
738 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
45 conjugacy classes — view the full character table.
| maximal subgroup of | S4×A4⋊S4, D3×S42, C3⋊S42, … |
| maximal normal subgroup of | D3×S42, S4×(C3⋊S4)⋊C3, C3×S4×C3⋊S4, … |
| maximal quotient of | S4×A4⋊S4, S4×#6258, #72222956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47852);