Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#985 |
| Fitting | C9×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C9 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C9 |
| Frattini, socle | C2×C3 | C2×C3×S4 |
| Fitting | C9×#41 | D3 |
| prime radical | C3×#985 | C3 |
| cosocle | C3×SL(2,𝔽3) | C22×C3 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4×C9 | |
127 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 127 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) = 144 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | #13146234, #13146237, C9×#5404, … |
| maximal normal subgroup of | #13146234, #13146237, C9×#5404, … |
| maximal quotient of | C4×C9×#226, C9×#5389, C4×C9×GL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2296);