Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C9 · Schur multiplier (H2) = C26 |
| direct product | C23×C9×C3⋊C8 |
| derived | C3 ⋊ C23×C8×C9 |
| Sylow | C3×C9 ⋊ C23×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4×C9 | D3 |
| commutator, hypocenter | C3 | C23×C8×C9 |
| Frattini | C3×C4 | C23×C3×D3 |
| Fitting | C23×C3×C4×C9 | C2 |
| socle, prime radical | C24×C32 | C3×C4 |
| cosocle | C32×C4 | C24×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C23×C4×C9 | |
1250 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
864 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 19588);