Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #3688⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C22×#3688⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | ((C22⋊C9)⋊C2)⋊C3 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C23×C3 |
| Frattini | C2×C3 | C22×C3×S4 |
| Fitting | C22×C32×Q8 | D3 |
| socle | C23×C3 | C3×S4 |
| prime radical | C23×C32 | S4 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C23 | |
566 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30468);