Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C3 |
| direct product | C3⋊D3×#882 |
| Fitting | C33×C22⋊C4 ⋊ C22 |
| Sylow | C33 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×C3⋊D3 |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C33×C22⋊C4 | C22 |
| hypercenter | C22⋊C4 | D3×C3⋊D3 |
| hypocenter | C33 | C2×#120 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
4492 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C3⋊D3×#120, #72258286 |
| maximal normal subgroup of | D3×C3⋊D3×#120, #72258286 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 41977);