Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C4⋊#4623 |
390 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 35 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^2=c,f^2=h,g^2=i,b^a=bc,e^a=e^2,f^a=fgi,g^a=gi,h^a=hi,f^e=gh,g^e=fg,h^e=i,i^e=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72209802, #72209803, #72209826, … |
| maximal normal subgroup of | #72210756, #72211388, #72213184, … |
| maximal quotient of | #72210756, #72210765, #72210769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085706);