Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #63419 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#10588303 |
| derived | C42⋊C9 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #63419 |
| commutator, hypocenter | C42⋊C9 | C3×C4 |
| Frattini | C23×C3 | C3×S4 |
| Fitting | C2×C32×C42 | D3 |
| socle, prime radical | C23×C32 | S4 |
| cosocle | C2×C42⋊C9 | C2×C3 |
| LCS | G ▷ C42⋊C9 | |
| UCS | C1 ◁ C2×C3 | |
202 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 202 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) = 72 |
108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2816);