Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22×#121 ⋊ D3 |
1106 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 63 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:
2^6,3,2^2|e^2=f,c^a=ce,e^a=ef,g^a=g^2,h^a=hi,c^b=cd,e^c=ef,h^g=i,i^g=hi52 conjugacy classes — view the full character table.
| maximal subgroup of | #72271405, #72271435, #72343920, … |
| maximal normal subgroup of | #72271435, #72343920, S4×#447 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087590);