Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #23467⋊C3 |
| Fitting, Sylow | #23467 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #120⋊C3 |
| commutator, cosocle, hypocenter | #543 | C22×C3 |
| Frattini | C24 | C22×A4 |
| Fitting | #23467 | C3 |
| socle | C22 | C2×#120⋊C3 |
| prime radical | #2029⋊C3 | C2 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C2×C4 | |
338 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|b^2=f,c^2=f,e^2=i,b^a=cefgi,c^a=bcghi,d^a=dh,g^a=h,h^a=gh,c^b=cf,d^b=dg,h^b=hi,d^c=dh,g^c=gi60 conjugacy classes — view the full character table.
| maximal subgroup of | #72227428, #72227992, #72268621, … |
| maximal normal subgroup of | C2×#23467⋊C3 |
| maximal quotient of | C2×#23467⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083908);