Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #4626 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C4 |
| direct product | C9×#4626 |
| derived | C42⋊C3 ⋊ C4×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #4626 |
| commutator, hypocenter | C42⋊C3 | C4×C9 |
| Frattini | C23×C3 | C3×S4 |
| Fitting | C9×#127 | D3 |
| socle | C22×C3 | C3×SL(2,ℤ/4) |
| prime radical, cosocle | C3×#127⋊C3 | C2×C3 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C9 | |
120 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2807);