Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#64201 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C22×C3 |
| Frattini | C2 | C3×D3×S4 |
| Fitting | C32×Q8 | C2×D3 |
| socle | C2×C32 | C2×S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
202 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
63 conjugacy classes — view the full character table.
| maximal subgroup of | #13180242, #13180244, #13180249, … |
| maximal normal subgroup of | #13180242, #13180244, #13180249, … |
| maximal quotient of | C3×#10593195, C3×#10593196, C3×#10593204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4133);