Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C22×C33 |
| direct product | C2×C32×C4×A4 |
| unambiguous semidirect product | (C23×C4)⋊C33 |
| unambiguous semidirect product | (C23×C3×C4)⋊C32 |
| Fitting | C23×C32×C4 ⋊ C3 |
| derived | C22 ⋊ C2×C33×C4 |
| Sylow | C23×C4 ⋊ C33 |
652 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 272 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
288 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C32×A4⋊C8, C32×#5409, C32×C4×SL(2,ℤ/4), … |
| maximal normal subgroup of | C2×C32×A4⋊C8, C32×#5409, C32×C4×SL(2,ℤ/4), … |
| maximal quotient of | C32×C42×A4, C32×A4×C22⋊C4, C32×A4×C4⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4601);