Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×#97 ⋊ D3 |
483 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 483 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) = 48 |
Polycyclic presentation:
2^6,3,2^2|b^2=e,c^2=d,d^2=f,e^2=f,b^a=be,e^a=ef,g^a=g^2,h^a=hi,c^b=cf,h^g=i,i^g=hi64 conjugacy classes — view the full character table.
| maximal subgroup of | #72241430, #72241441, S4×#444, … |
| maximal normal subgroup of | #72241430, #72241441, S4×#444 |
| maximal quotient of | #72342714 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087243);