Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C3×C64 ⋊ C2 |
| Sylow | C3 ⋊ #13737 |
| subgroup | G/H | |
|---|---|---|
| center | C64 | C2×D3 |
| commutator, socle | C2×C3 | C22×C32 |
| Frattini | C32 | C22×D3 |
| Fitting | C2×C3×C64 | C2 |
| prime radical | #235 | C16 |
| cosocle | C3×C32 | C23 |
| hypercenter | C2×C64 | D3 |
| hypocenter | C3 | #13737 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C64 ◁ C2×C64 | |
128 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=d,d^2=e,e^2=f,f^2=g,g^2=h,c^b=ch,i^c=i^2288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 90606);