Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C32 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #13737⋊C3 |
| Fitting, Sylow | #13737 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C64 | A4 |
| commutator, hypocenter | Q8 | C3×C32 |
| Frattini | C32 | C2×A4 |
| Fitting | #13737 | C3 |
| socle | C2 | A4×C32 |
| prime radical | #41⋊C3 | C16 |
| cosocle | #2258 | C2×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C64 | |
43 conjugacy classes of subgroups.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
3,2^8|b^2=e,e^2=f,f^2=g,g^2=h,h^2=i,c^a=cdhi,d^a=c,d^c=di224 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#13737⋊C3 |
| maximal normal subgroup of | C3×#13737⋊C3 |
| maximal quotient of | SL(2,𝔽3)×C64, C3×#13737⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083577);