Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | D9×GL(2,𝔽3) |
| Fitting | Q8×C9 ⋊ C2×D3 |
| derived | C9×SL(2,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9×S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C22 |
| Frattini, socle | C2×C3 | D3×S4 |
| Fitting | Q8×C9 | C2×D3 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
167 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D9×GL(2,𝔽3), D9×#983, #13164125, … |
| maximal normal subgroup of | C2×D9×GL(2,𝔽3), D9×#983, #13164125, … |
| maximal quotient of | D9×#859, Dic9×GL(2,𝔽3), #13146228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2204);