Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #10593244 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10593244 |
| Fitting | C24×C32 ⋊ C2×D3 |
| derived | C3×C22⋊A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #10593244 |
| commutator, cosocle, hypocenter | C3×C22⋊A4 | C22×C3 |
| Frattini | C22 | C3×D3×S4 |
| Fitting | C24×C32 | C2×D3 |
| socle | C22×C32 | C2×S4 |
| LCS | G ▷ C3×C22⋊A4 | |
| UCS | C1 ◁ C3 | |
504 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 504 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) = 24 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | #72210357, C32×#10593244, C3×#13166068, … |
| maximal normal subgroup of | #72210357, C32×#10593244, C3×D3×#1020⋊C2, … |
| maximal quotient of | C32×#10593244, C3×#13166068, C3×#13165994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31983);