Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | AΣL(1,𝔽27) ⋊ Q8 |
102 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 102 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^4|a^2=c,b^2=c,b^a=bc,e^a=e^2,f^a=f^2g,d^b=d^2,f^b=f^2,g^b=g^2,e^d=ef,f^e=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | #4174⋊Q8, #4183⋊Q8, #4175⋊Q8, … |
| maximal normal subgroup of | #4174⋊Q8, #4183⋊Q8, #4175⋊Q8, … |
| maximal quotient of | #13046484, #13046485, #13046486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 71);