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×Dic3×S4 |
| unambiguous semidirect product | (C3×S4)⋊(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×C32×S4 | C2 |
| cosocle | C2×C3×A4 | C22×C3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C2×C3 | |
289 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic3×S4, S4×#624, #13180282, … |
| maximal normal subgroup of | D3×Dic3×S4, S4×#624, #13180282, … |
| maximal quotient of | C3×C3⋊C8×S4, C3×#10593175, C3×Dic3×#226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4137);