Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C22×C3×D3×Dic3 |
| derived | C32 ⋊ C23×C3×C4 |
| Sylow | C33 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | D32 |
| commutator, hypocenter | C32 | C23×C3×C4 |
| Frattini | C2 | C22×C3×D32 |
| Fitting, socle | C23×C33 | C22 |
| prime radical | C23×C32×D3 | C2 |
| cosocle | C2×C32 | C24×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C23×C3 | |
1218 conjugacy classes of subgroups; only the 456 normal subgroups are shown.
This group has 1218 conjugacy classes of subgroups — too many to draw. Only its 456 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
216 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63579, Dic3×(C2×D3)⋊Dic3, #13156695, … |
| maximal normal subgroup of | D3×#63579, Dic3×(C2×D3)⋊Dic3, #13156695, … |
| maximal quotient of | C2×C3×Dic3×Dic6, C2×C3×#63843, C3×#10594818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4516);