Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C3⋊(A4⋊C8) |
| unambiguous semidirect product | C3⋊(C3×A4⋊C8) |
| unambiguous semidirect product | A4⋊(C3×C3⋊C8) |
| derived | C3×A4 ⋊ C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | C3⋊S4 |
| commutator, hypocenter | C3×A4 | C3×C8 |
| Frattini | C4 | C3×C3⋊S4 |
| Fitting | C22×C32×C4 | D3 |
| socle | C23×C32 | Dic3 |
| prime radical | C2×C32×A4 | C4 |
| cosocle | C3×C4×A4 | C2×C3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C3×C4 | |
191 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 50 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×C3⋊C8×S4, C3×#10593175, C3×D3×A4⋊C8, … |
| maximal normal subgroup of | C3×C3⋊C8×S4, C3×#10593175, C3×D3×A4⋊C8, … |
| maximal quotient of | C3×C3⋊(SL(2,𝔽3)⋊C8), C3×C3⋊(A4⋊C16), C3×C3⋊#4628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3010);