Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| direct product | D3×C3⋊#226 |
| unambiguous semidirect product | (C3×D3)⋊#226 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×C3⋊S4 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C22 |
| Fitting | C32×Q8 | C2×D3 |
| socle | C2×C32 | C2×S4 |
| prime radical | C3×D3×SL(2,𝔽3) | C2 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
251 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#64196, D3×#64197, #13180256, … |
| maximal normal subgroup of | D3×#64196, D3×#64197, #13180256, … |
| maximal quotient of | Dic3×C3⋊#226, #13166017, #13166024, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4162);