Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #4625⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (#1020⋊C2)⋊C4 |
| Fitting | #2375 ⋊ D3 |
372 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^4,3,2^4|b^2=c,b^a=bd,e^a=e^2,f^a=fg,h^a=hi,f^b=fi,g^b=ghi,f^d=fh,g^d=gi,f^e=g,g^e=fg,h^e=i,i^e=hi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72236332, #72236358, C4×#4625⋊C2, … |
| maximal normal subgroup of | #72236332, C4×#4625⋊C2 |
| maximal quotient of | #72253396, #72253567 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085963);