Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #11337⋊C3 |
| Fitting, Sylow | #11337 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #127⋊C3 |
| commutator, hypocenter | #338 | C3×C4 |
| Frattini | C23×C4 | C2×A4 |
| Fitting | #11337 | C3 |
| socle | C22 | #392⋊C3 |
| prime radical | C2×#338⋊C3 | C2 |
| cosocle | C2×#338 | C2×C3 |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C2×C4 | |
156 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bcf,c^a=cdfghi,d^a=cefghi,e^a=efgi,f^a=e,c^b=ce,d^b=df,e^b=ei,f^b=fi,d^c=dg,f^c=fi,e^d=ei60 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11337⋊C3, C4×#2017⋊C3, #72214726 |
| maximal normal subgroup of | C2×#11337⋊C3, C4×#2017⋊C3, #72214726 |
| maximal quotient of | #72211239, C2×#11337⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083551);