Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×A42 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | Dic3×A42 |
| unambiguous semidirect product | C3⋊(C4×A42) |
| unambiguous semidirect product | (C22×Dic3×A4)⋊C3 |
| unambiguous semidirect product | (Dic3×C22⋊A4)⋊C3 |
| derived | C24×C3 ⋊ C32×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×A42 |
| commutator, hypocenter | C24×C3 | C32×C4 |
| Fitting, socle, cosocle | C25×C3 | C2×C32 |
| prime radical | C2×C3×A42 | C2 |
| LCS | G ▷ C24×C3 | |
| UCS | C1 ◁ C2 | |
490 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 55 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.
| maximal subgroup of | #72242698, C4×D3×A42 |
| maximal normal subgroup of | C4×D3×A42 |
| maximal quotient of | #72242698 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46410);