Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24×C3 |
| direct product | C22×C3×C3⋊S4 |
| Fitting, socle | C24×C32 ⋊ D3 |
| derived | C3×A4 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | C3⋊S4 |
| commutator, cosocle, hypocenter | C3×A4 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C32 | D3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C22×C3 | |
751 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 751 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#10593242, #13166065, #13166114, … |
| maximal normal subgroup of | C3×#10593242, #13166065, #13166114, … |
| maximal quotient of | C2×C3×#64259, C2×C3×C4×C3⋊S4, C2×C3×C4⋊(C3⋊S4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4706);