Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C27×C3 |
| direct product | C23×C32×S4 |
| unambiguous semidirect product | A4⋊(C24×C32) |
| unambiguous semidirect product | (C2×A4)⋊(C23×C32) |
| Fitting, socle | C25×C32 ⋊ D3 |
| derived | A4 ⋊ C24×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C32 | S4 |
| commutator, cosocle, hypocenter | A4 | C24×C32 |
| Frattini | C1 | G |
| Fitting, socle | C25×C32 | D3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C23×C32 | |
4462 conjugacy classes of subgroups; only the 594 normal subgroups are shown.
This group has 4462 conjugacy classes of subgroups — too many to draw. Only its 594 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
360 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×S4×#1094 |
| maximal normal subgroup of | C2×S4×#1094 |
| maximal quotient of | C23×C3×((C22⋊C9)⋊C2)⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47916);