Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×D8×S4 |
| Fitting | C23×D8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×S4 |
| commutator, cosocle | C4×A4 | C24 |
| Frattini | C4 | C23×S4 |
| Fitting | C23×D8 | D3 |
| socle | C24 | D3×D4 |
| hypercenter | C2×D8 | S4 |
| hypocenter | A4 | C22×D8 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
1889 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 1889 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | #72226317, C2×S4×AGL(1,ℤ/8), #72240991, … |
| maximal normal subgroup of | #72226317, C2×S4×AGL(1,ℤ/8), S4×#509, … |
| maximal quotient of | #72226317, S4×#509, D8×GL(2,ℤ/4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089270);