Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C3 |
| direct product | #67×C3⋊S4 |
| Fitting | C24×C32 ⋊ C2×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C3⋊S4 |
| commutator, cosocle | C2×C32×A4 | C23 |
| Fitting | C24×C32 | C2×D3 |
| socle | C23×C32 | C22×D3 |
| hypercenter | C22 | D3×C3⋊S4 |
| hypocenter | C32×A4 | C2×D4 |
| LCS | G ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1310 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1310 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
81 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊S42, #67×(C3⋊S4)⋊C3, #72222973, … |
| maximal normal subgroup of | #67×(C3⋊S4)⋊C3, #72222977, D3×#67×S4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46903);