Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊D3×#857 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| direct product | C3⋊D3×#857 |
| Fitting | C32×C42 ⋊ C2×D3 |
| derived | C32×C42⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×C42⋊C3 | C22 |
| Frattini | C22 | C3⋊D3×S4 |
| Fitting | C32×C42 | C2×D3 |
| socle | C22×C32 | C2×S4 |
| LCS | G ▷ C32×C42⋊C3 | |
| UCS | C1 | |
552 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31984);