Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | SL(2,𝔽3)⋊C32 |
| derived | SL(2,𝔽3) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C16 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C32 |
| Fitting | Q8×C16 | D3 |
| socle | C22 | A4⋊C16 |
| prime radical | C2×SL(2,𝔽3) | C16 |
| cosocle | C16×SL(2,𝔽3) | C2 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C16 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
2^5,3,2^3|a^2=b,b^2=c,c^2=d,d^2=e,g^2=i,h^2=i,f^a=f^2,g^a=gh,h^a=hi,g^f=h,h^f=ghi,h^g=hi128 conjugacy classes — view the full character table.
| maximal subgroup of | C3×SL(2,𝔽3)⋊C32, C32×GL(2,𝔽3), #72219100 |
| maximal normal subgroup of | C3×SL(2,𝔽3)⋊C32, C32×GL(2,𝔽3) |
| maximal quotient of | C3×SL(2,𝔽3)⋊C32, #72219100 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085340);