Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #126⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C9×#383⋊C3 |
| Fitting | C9×#383 ⋊ C3 |
| derived | C42 ⋊ C3×C4×C9 |
| Sylow | #383 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | #126⋊C3 |
| commutator, hypocenter | C42 | C3×C4×C9 |
| Frattini, socle | C23×C3 | C2×C3×A4 |
| Fitting | C9×#383 | C3 |
| prime radical | C2×C3×C42⋊C3 | C2×C3 |
| cosocle | C2×C3×C42 | C2×C32 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C2×C9 | |
150 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7251);