Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C32 |
| unambiguous semidirect product | C4⋊(C3⋊((C3⋊D3)⋊C3)) |
| Fitting | C3×C4×H(3,𝔽3) ⋊ C2 |
| Sylow | C3×H(3,𝔽3) ⋊ D4 |
253 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^2,3,2,3^3|b^2=d,b^a=bd,e^a=e^2,f^a=f^2,g^a=g^2,e^c=eg75 conjugacy classes — view the full character table.
| maximal subgroup of | #88797⋊C3, #88791⋊C3, #88790⋊C3, … |
| maximal normal subgroup of | #88797⋊C3, #88791⋊C3, #88790⋊C3, … |
| maximal quotient of | #13047895, #13047898, #13047899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 478);