Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #14601⋊C3 |
| Fitting, Sylow | #14601 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #5450 |
| commutator, hypocenter | C22×Q8 | C2×C3×C4 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #14601 | C3 |
| prime radical | C2×C22⋊SL(2,𝔽3) | C22 |
| cosocle | C23×Q8 | C22×C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C22 | |
275 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=fh,d^2=fhi,e^2=fg,b^a=defg,c^a=bcdefi,d^a=beh,e^a=cf,f^a=h,g^a=f,h^a=g,i^a=ghi,c^b=cf,d^b=dg,e^b=efg,d^c=dh,e^c=ef48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083633);