Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #11380⋊C3 |
| Fitting, Sylow | #11380 ⋊ C3 |
| derived | C22×Q8 ⋊ C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #120⋊C3 |
| commutator, hypocenter | C22×Q8 | C3×C8 |
| Frattini | C23×C4 | C2×A4 |
| Fitting | #11380 | C3 |
| socle | C24 | C4×A4 |
| prime radical | C2×C22⋊SL(2,𝔽3) | C4 |
| cosocle | C22×C4×Q8 | C2×C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2×C4 | |
261 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=g,d^2=g,h^2=i,c^a=cdef,d^a=cfg,e^a=ef,f^a=e,c^b=ce,d^b=df,d^c=dg72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083553);