Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C42 |
| unambiguous semidirect product | #16248⋊C3 |
| Fitting, derived, Sylow | #16248 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #5464 |
| commutator, Fitting, cosocle, hypocenter | #16248 | C3 |
| Frattini, socle | C24 | C22⋊A4 |
| LCS | G ▷ #16248 | |
| UCS | C1 ◁ C22 | |
280 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|b^2=fg,c^2=fgi,d^2=hi,e^2=fgh,b^a=cdh,c^a=bcefghi,d^a=de,e^a=dh,f^a=gh,g^a=fh,h^a=fghi,i^a=h,c^b=cf,d^b=dg,e^b=ei,d^c=dh,e^c=ef30 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083698);