Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#3407 ⋊ C2 |
| Sylow | C3 ⋊ #33392 |
1154 conjugacy classes of subgroups; only the 351 normal subgroups are shown.
This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 351 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:
2^8,3|a^2=f,e^2=g,f^2=h,g^2=h,b^a=be,e^a=eg,g^a=gh,i^a=i^2,e^b=egh,f^b=fh,g^b=gh,i^b=i^2,d^c=dh126 conjugacy classes — view the full character table.
| maximal subgroup of | #72222830, D3×#33392, #72258593, … |
| maximal normal subgroup of | D3×#33392, #72258593, #72258653, … |
| maximal quotient of | #72222830 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 322535);