Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×2−1+4 ⋊ D3 |
1050 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^5,3,2^3|e^2=i,g^2=i,h^2=i,b^a=be,c^a=cd,e^a=ei,f^a=f^2,g^a=gh,h^a=hi,e^b=ei,g^f=h,h^f=ghi,h^g=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72227580 |
| maximal quotient of | #72227580, #72267941, #72271322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088332);