Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | S4×#135 |
| Fitting | C22×#135 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×S4 |
| commutator, cosocle | C4×A4 | C24 |
| Frattini | C4 | C23×S4 |
| Fitting | C22×#135 | D3 |
| socle | C23 | C2×D3×D4 |
| hypercenter | #135 | S4 |
| hypocenter | A4 | C2×#135 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #135 | |
1447 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 1447 conjugacy classes of subgroups — too many to draw. Only its 121 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 | S4×#509, S4×#360, S4×#444, … |
| maximal normal subgroup of | S4×#509, S4×#360, S4×#444, … |
| maximal quotient of | C4×D8×S4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089282);