Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | S4×#123 |
| Fitting | C22×#123 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×S4 |
| commutator, cosocle | C22×A4 | C24 |
| Fitting | C22×#123 | D3 |
| socle | C24 | C23×D3 |
| prime radical | C2×D4×S4 | C2 |
| hypercenter | #123 | S4 |
| hypocenter | A4 | C2×#123 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ #123 | |
1911 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 1911 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 | S4×#525, #72341575 |
| maximal normal subgroup of | S4×#525 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088769);