Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#1010⋊C3 |
| unambiguous semidirect product | #1010⋊C32 |
| unambiguous semidirect product | (C3×#1010)⋊C3 |
| Fitting | C32×#41 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D3×A4 |
| commutator, cosocle, hypocenter | C3×Q8 | C22×C32 |
| Frattini | C2 | C2×C3×D3×A4 |
| Fitting | C32×#41 | C2×C3 |
| socle | C2×C32 | C22×A4 |
| hypercenter | C3×C4 | D3×A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
281 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 281 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
99 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#10593229, C3×#10593230, C3×#10593231, … |
| maximal normal subgroup of | C3×#10593229, C3×#10593230, C3×#10593231, … |
| maximal quotient of | C3×#5570⋊C3, C3×SL(2,𝔽3)×D12, C3×#5572⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4449);