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 | #5450⋊C4 |
332 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^4,3,2^4|a^2=d,b^a=bc,e^a=e^2,f^a=fg,h^a=hi,f^b=fi,g^b=ghi,f^c=fh,g^c=gi,f^e=g,g^e=fg,h^e=i,i^e=hi32 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4625⋊C2, #72243028, #72247747, … |
| maximal normal subgroup of | C4×#4625⋊C2, #72258617, C2×#5450⋊C4 |
| maximal quotient of | #72248152, #72253380, C2×#5450⋊C4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085908);