Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | A4⋊#564 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×S4 |
| commutator, prime radical, cosocle | C22×A4 | C24 |
| Fitting | C22×#125 | D3 |
| socle | C24 | C23×D3 |
| hypercenter | #125 | S4 |
| hypocenter | A4 | #564 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ #125 | |
925 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 925 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3,2^2|a^2=f,b^2=e,c^2=ef,d^2=f,c^a=cf,d^a=de,g^a=g^2,h^a=hi,c^b=cef,d^b=df,d^c=df,h^g=i,i^g=hi52 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088918);