Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×#107 ⋊ D3 |
291 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3,2^2|b^2=c,d^2=ef,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^2,h^a=hi,d^b=de,e^b=ef,h^g=i,i^g=hi58 conjugacy classes — view the full character table.
| maximal subgroup of | #72342748, S4×#357, #72342792 |
| maximal normal subgroup of | #72342748, S4×#357, #72342792 |
| maximal quotient of | #72342756, #72342805 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085948);