Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3×C4 |
| direct product | C3⋊D3×#874 |
| Fitting | C33×C42 ⋊ C22 |
| Sylow | C33 ⋊ C2×#127 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×C3⋊D3 |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C33×C42 | C22 |
| hypercenter | C42 | D3×C3⋊D3 |
| hypocenter | C33 | C2×#127 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C42 | |
3420 conjugacy classes of subgroups; only the 364 normal subgroups are shown.
This group has 3420 conjugacy classes of subgroups — too many to draw. Only its 364 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
180 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C3⋊D3×#127, #72220384 |
| maximal normal subgroup of | D3×C3⋊D3×#127, #72220384 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 41461);