Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
201 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3,2^2|a^2=d,b^2=cef,c^2=ef,e^2=f,b^a=bc,c^a=ce,e^a=ef,g^a=g^2,h^a=hi,h^g=i,i^g=hi76 conjugacy classes — view the full character table.
| maximal subgroup of | #72342690, #72342785, S4×#367, … |
| maximal normal subgroup of | #72342690, #72342785, S4×#367, … |
| maximal quotient of | #72342820 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085918);