Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| direct product | C3⋊D3×#226 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊D3×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 | |
274 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | #157×#226, #13180184, #13180235, … |
| maximal normal subgroup of | #157×#226, #13180184, #13180235, … |
| maximal quotient of | C3⋊Dic3×#226, #13165943, #13165951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4147);