Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #13749⋊C3 |
| Fitting, Sylow | #13749 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C42⋊C3 |
| commutator, hypocenter | #95 | C2×C3×C4 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #13749 | C3 |
| prime radical | C22×#95⋊C3 | C2 |
| cosocle | C2×#95 | C22×C3 |
| hypercenter | C22⋊C4 | C42⋊C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
503 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=g,c^2=h,d^2=i,b^a=bcfgi,c^a=cde,d^a=cefgh,g^a=fgh,h^a=fhi,i^a=h,c^b=cf,e^b=ef,d^c=df72 conjugacy classes — view the full character table.
| maximal subgroup of | #72222668, #72222701, #72227439, … |
| maximal normal subgroup of | #72222668, #72222701, #72227450, … |
| maximal quotient of | #72222564, C2×#13749⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083584);