Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #857⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#857⋊C2 |
| unambiguous semidirect product | #857⋊(C2×C9) |
| Fitting | C9×#127 ⋊ D3 |
| derived | C42⋊C3 ⋊ C22×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #857⋊C2 |
| commutator, hypocenter | C42⋊C3 | C22×C9 |
| Frattini, socle | C22×C3 | C2×C3×S4 |
| Fitting | C9×#127 | D3 |
| prime radical | C3×#857⋊C2 | C3 |
| cosocle | C3×C42⋊C3 | C22×C3 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C9 | |
261 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12824);