Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Sylow | C3×H(3,𝔽3) ⋊ Q8 |
291 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 93 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,d^a=d^2,e^a=e^2,f^a=f^2,f^e=fg75 conjugacy classes — view the full character table.
| maximal subgroup of | #13047681, #13047682, #13047683, … |
| maximal normal subgroup of | #13047681, #13047682, #13047683, … |
| maximal quotient of | #13048016, C4⋊(C3⋊#1084), C2×#10619988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 507);