Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×#63422⋊C3 |
| unambiguous semidirect product | (C2×#63422)⋊C3 |
| derived | Q8⋊C9 ⋊ C2×C3×C4 |
322 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
116 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10589531⋊C3, #72245649, #72245981, … |
| maximal normal subgroup of | C2×#10589531⋊C3, #72245649, #72245981 |
| maximal quotient of | C2×#72252951 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12972);