Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D3×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | D3×C8×D9 |
| Fitting | C3×C8×C9 ⋊ C22 |
| derived, Sylow | C3×C9 ⋊ C22×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D3×D9 |
| commutator, hypocenter | C3×C9 | C22×C8 |
| Frattini | C3×C4 | C2×D32 |
| Fitting | C3×C8×C9 | C22 |
| socle | C2×C32 | C2×C4×D3 |
| prime radical | C2×D3×D9 | C4 |
| cosocle | C3×C4×C9 | C23 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C8 | |
250 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 79 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 | D3×C16×D9, D9×#798, D3×#63357, … |
| maximal normal subgroup of | D3×C16×D9, D9×#798, D3×#63357, … |
| maximal quotient of | D3×C16×D9, D9×#798, D3×#63357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 765);