Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #17230⋊C3 |
| Fitting, Sylow | #17230 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #560⋊C3 |
| commutator, cosocle | C2×#95 | C22×C3 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #17230 | C3 |
| prime radical | C2×#95⋊C3 | C22 |
| hypocenter | #95 | C3×D4 |
| LCS | G ▷ C2×#95 ▷ #95 | |
| UCS | C1 ◁ C22 | |
225 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 225 conjugacy classes of subgroups — too many to draw. Only its 28 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=fgh,c^2=g,d^2=fg,e^2=gh,b^a=bcdg,c^a=dgh,d^a=cd,e^a=cefgh,f^a=gh,g^a=fg,i^a=fghi,c^b=cf,d^b=dg,e^b=ei,d^c=dh,e^c=efg,e^d=ef42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083715);