Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D32×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×D32×A4 |
| unambiguous semidirect product | (C3×D3)⋊(C22×A4) |
| unambiguous semidirect product | (C23×D32)⋊C3 |
| Fitting, socle | C23×C32 ⋊ C22×C3 |
| derived | C22×C32 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D32×A4 |
| commutator, cosocle, hypocenter | C22×C32 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | C22×C3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C2 | |
799 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 799 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | D32⋊SL(2,ℤ/4), A4×D32⋊C4, D32×SL(2,ℤ/4), … |
| maximal normal subgroup of | D32⋊SL(2,ℤ/4), A4×D32⋊C4, D32×SL(2,ℤ/4), … |
| maximal quotient of | D3×A4×Dic6, A4×#3795, A4×#3796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4694);