Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C211 |
| direct product | C23×D4×A4 |
| unambiguous semidirect product | C4⋊(C24×A4) |
| Fitting, Sylow | C25×D4 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×A4 |
| commutator, cosocle | C23 | C25×C3 |
| Frattini | C2 | C25×A4 |
| Fitting | C25×D4 | C3 |
| socle | C26 | C22×C3 |
| hypercenter | C23×D4 | A4 |
| hypocenter | C22 | C23×C3×D4 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C24 ◁ C23×D4 | |
20803 conjugacy classes of subgroups; only the 1275 normal subgroups are shown.
This group has 20803 conjugacy classes of subgroups — too many to draw. Only its 1275 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×A42, C2×D4×GL(2,ℤ/4), C22×#85489, … |
| maximal normal subgroup of | C2×D4×A42, C2×D4×GL(2,ℤ/4), C22×#85489, … |
| maximal quotient of | #72226247, C22×A4×AGL(1,ℤ/8), C22×A4×#127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085288);