Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C3 |
| direct product | D3×#64162 |
| Fitting | C24×C33 ⋊ C22 |
| Sylow | C33 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×C3⋊D3 |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C24×C33 | C22 |
| hypercenter | C24 | D3×C3⋊D3 |
| hypocenter | C33 | C2×#120 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C24 | |
4492 conjugacy classes of subgroups; only the 476 normal subgroups are shown.
This group has 4492 conjugacy classes of subgroups — too many to draw. Only its 476 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
234 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C3⋊D3×#120, D3×C3⋊(C3⋊#1020), D3×#10591725 |
| maximal normal subgroup of | D3×C3⋊D3×#120, D3×#10591725 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 44168);