Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #10596770 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#10596770 |
| Fitting, socle | C24×C3 ⋊ #157 |
| derived | A42 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #10596770 |
| commutator, hypocenter | A42 | C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3 | #157 |
| prime radical | C3×A4⋊S4 | C2 |
| cosocle | A4⋊S4 | C2×C3 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C3 | |
151 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
39 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#10596770⋊C2, #72275397, D3×#10596770, … |
| maximal normal subgroup of | C3×#10596770⋊C2, #72275397, D3×#10596770, … |
| maximal quotient of | C2×C3×#10596770 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47849);