Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22×#126 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×S4 |
| commutator, cosocle | C22×A4 | C24 |
| Fitting | C22×#126 | D3 |
| socle | C24 | C23×D3 |
| prime radical | C2×GL(2,ℤ/4) | C22 |
| hypercenter | #126 | S4 |
| hypocenter | A4 | #563 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ #126 | |
1054 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3,2^2|b^2=f,d^2=e,c^a=cf,d^a=def,g^a=g^2,h^a=hi,c^b=ce,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, 1088912);