Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C64 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | A4⋊C64 |
| derived | A4 ⋊ C64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | S4 |
| commutator, hypocenter | A4 | C64 |
| Fitting | C22×C32 | D3 |
| socle | C23 | C3⋊C32 |
| prime radical | C2×A4 | C32 |
| cosocle | A4×C32 | C2 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C32 | |
47 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3,2^2|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,g^a=g^2,h^a=hi,h^g=i,i^g=hi160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085351);