Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×M16 ⋊ D3 |
516 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^5,3,2^3|c^2=d,d^2=e,g^2=i,h^2=i,c^a=cd,d^a=de,f^a=f^2,g^a=gh,h^a=hi,c^b=ce,g^f=h,h^f=ghi,h^g=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72220196, AGL(1,ℤ/8)×GL(2,𝔽3) |
| maximal normal subgroup of | AGL(1,ℤ/8)×GL(2,𝔽3) |
| maximal quotient of | #72220196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086558);