Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C42 · Schur multiplier (H2) = C22×C3×C4 |
| direct product | C32×C42×D3 |
| Fitting | C33×C42 ⋊ C2 |
| derived | C3 ⋊ C2×C32×C42 |
| Sylow | C33 ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C42 | D3 |
| commutator, hypocenter | C3 | C2×C32×C42 |
| Frattini | C22 | C22×C32×D3 |
| Fitting | C33×C42 | C2 |
| socle | C22×C33 | C23 |
| prime radical | C22×C32×D3 | C22 |
| cosocle | C22×C3 | C23×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C32×C42 | |
768 conjugacy classes of subgroups; only the 414 normal subgroups are shown.
This group has 768 conjugacy classes of subgroups — too many to draw. Only its 414 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
432 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×#10590261, C3×#10590248, C3×#10590262, … |
| maximal normal subgroup of | C3×#10590261, C3×#10590248, C3×#10590262, … |
| maximal quotient of | C32×#4666, C32×#4648, C32×#4649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3707);