Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
1289 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 1289 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^4,3,2^4|a^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=fg,h^a=hi,c^b=cd,f^e=g,g^e=fg,h^e=i,i^e=hi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72212876, #72226632, #72228082, … |
| maximal normal subgroup of | SD16×#1020, #72234216, #72241506, … |
| maximal quotient of | #72238100, #72242986, C2×#11715696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090077);