Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | D3×#3688 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | C3×Q8⋊C9 | C22 |
| Frattini | C2×C3 | D3×S4 |
| Fitting | C32×Q8 | C2×D3 |
| socle | C2×C32 | C2×S4 |
| prime radical | C2×C3×D3 | S4 |
| LCS | G ▷ C3×Q8⋊C9 | |
| UCS | C1 ◁ C2 | |
143 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
45 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×#3688, #13164188, #13164189, … |
| maximal normal subgroup of | C2×D3×#3688, #13164188, #13164189, … |
| maximal quotient of | Dic3×#3688, #13146301, #13146309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2216);