Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #4174⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #4175⋊C4 |
| derived, Sylow | AΣL(1,𝔽27) ⋊ C2×C4 |
134 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^3,3^4|a^2=c,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=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | #4175⋊Q8, #13047115, #13047116, … |
| maximal normal subgroup of | #4175⋊Q8, #13047115, #13047116, … |
| maximal quotient of | #4175⋊C8, #13046412, #13046480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 67);