Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C25×C3×C4 |
| direct product | C22×C3×C4×C3⋊Dic3 |
| derived | C32 ⋊ C22×C3×C42 |
| Sylow | C33 ⋊ C22×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3×C4 | C3⋊D3 |
| commutator, hypocenter | C32 | C22×C3×C42 |
| Frattini | C22 | C23×C3×C3⋊D3 |
| Fitting | C23×C33×C4 | C2 |
| socle, prime radical | C24×C33 | C22 |
| cosocle | C22×C32 | C24×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C23×C3×C4 | |
3932 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
576 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 45596);