Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C4⋊#5936 |
1580 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1580 conjugacy classes of subgroups — too many to draw. Only its 51 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:
2^4,3,2^4|a^2=d,b^2=c,b^a=bc,e^a=e^2,f^a=fg,h^a=hi,f^e=g,g^e=fg,h^e=i,i^e=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | C4⋊C4×#1020, #72238097, #72238100, … |
| maximal normal subgroup of | C4⋊C4×#1020, #72238100, #72253057, … |
| maximal quotient of | #72227311, #72227317, #72253057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089951);