Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D3×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×D3×SL(2,ℤ/4) |
| unambiguous semidirect product | (D3×A4)⋊(C3×C4) |
| derived | C3×A4 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×S4 |
| commutator, hypocenter | C3×A4 | C2×C3×C4 |
| Frattini | C2 | C3×D3×S4 |
| Fitting, socle | C23×C32 | C2×D3 |
| prime radical | C2×C3×D3×A4 | C2 |
| cosocle | C2×C3×A4 | C22×C3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C2×C3 | |
311 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#64204, D3×(C3⋊S4)⋊C4, #13180295, … |
| maximal normal subgroup of | D3×#64204, D3×(C3⋊S4)⋊C4, #13180295, … |
| maximal quotient of | C3×D3×A4⋊C8, C3×#10593182, C3×#10593191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4140);