Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C33 |
| direct product | D4×C4⋊(C3⋊(C3⋊D3)) |
| Fitting | C33×C4×D4 ⋊ C2 |
| Sylow | C33 ⋊ D42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C33×C4×D4 | C2 |
| hypercenter | C4×D4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | D42 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
5992 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
285 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 45740);