Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#64264 |
| Fitting | C2×C32×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C22×C3 |
| Frattini | C2 | C2×C3×C3⋊S4 |
| Fitting | C2×C32×Q8 | D3 |
| socle | C2×C32 | C2×S4 |
| hypercenter | C22×C3 | C3⋊S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
268 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
81 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#10596435, C3×#10596439, C3×D3×#983, … |
| maximal normal subgroup of | C3×#10596435, C3×#10596439, C3×D3×#983, … |
| maximal quotient of | C3×#10593622, C3×#10593623, C3×#10593628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4424);