Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #131⋊SL(2,𝔽3) |
| Fitting, Sylow | Q8×#131 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | C22⋊A4 |
| commutator, hypocenter | Q82 | C3×C4 |
| Frattini | C2×C4 | C2×C22⋊A4 |
| Fitting | Q8×#131 | C3 |
| socle | C22 | C4×C22⋊A4 |
| prime radical | #41⋊SL(2,𝔽3) | C2 |
| cosocle | Q8×#41 | C2×C3 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C2×C8 | |
377 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 377 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=g,d^2=gh,h^2=i,b^a=bc,c^a=bdhi,e^a=f,f^a=efgh,c^b=cg,f^e=fi76 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084662);