Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #13464⋊C3 |
| Fitting, Sylow | #13464 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | #2811 | C2×C3 |
| Frattini | C22×Q8 | C2×A4 |
| Fitting | #13464 | C3 |
| prime radical | #73291 | C2 |
| hypercenter | C23 | #120⋊C3 |
| LCS | G ▷ #2811 | |
| UCS | C1 ◁ C22 ◁ C23 | |
131 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 131 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|b^2=e,c^2=e,d^2=e,f^2=i,g^2=i,b^a=cei,c^a=bch,d^a=dfghi,f^a=gh,g^a=fghi,c^b=ce,d^b=df,f^b=fi,g^b=ghi,d^c=dg,f^c=fh,g^c=gi,f^d=fi,g^d=gi,h^d=hi,g^f=gi37 conjugacy classes — view the full character table.
| maximal subgroup of | #72268097 |
| maximal normal subgroup of | #72268097 |
| maximal quotient of | #72268602 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083574);