Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C22 |
| direct product | C2×C9×SL(2,ℤ/4) |
| unambiguous semidirect product | (C2×A4)⋊(C4×C9) |
| derived | A4 ⋊ C2×C4×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C9 | S4 |
| commutator, hypocenter | A4 | C2×C4×C9 |
| Frattini | C2×C3 | C2×C3×S4 |
| Fitting | C24×C9 | D3 |
| socle | C24×C3 | C3×D3 |
| prime radical | C22×C3×A4 | C2×C3 |
| cosocle | C2×C3×A4 | C22×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C22×C9 | |
209 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
180 conjugacy classes — view the full character table.
| maximal subgroup of | Dic9×SL(2,ℤ/4), (C2×D9)⋊SL(2,ℤ/4), #13146258, … |
| maximal normal subgroup of | Dic9×SL(2,ℤ/4), (C2×D9)⋊SL(2,ℤ/4), #13146258, … |
| maximal quotient of | C2×C9×A4⋊C8, C9×#5409, C4×C9×SL(2,ℤ/4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2298);