Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23 |
| direct product | A4×#504 |
| Fitting, Sylow | C22×#504 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C16 | C22×A4 |
| commutator, socle | C23 | C22×C3×C8 |
| Frattini | C8 | C23×A4 |
| Fitting | C22×#504 | C3 |
| prime radical | A4×#41 | C4 |
| cosocle | C22×C8 | C23×C3 |
| hypercenter | #504 | A4 |
| hypocenter | C22 | C3×#504 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C16 ◁ #504 | |
493 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 493 conjugacy classes of subgroups — too many to draw. Only its 117 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 | A4×#2171, S4×#504, #72343063 |
| maximal normal subgroup of | A4×#2171, S4×#504, #72343063 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084935);