Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32⋊C9 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ((C3⋊D3)⋊C9)⋊C2 |
| commutator, Fitting, cosocle | C2×C32⋊C9 | C22 |
| Frattini, socle | C2×C32 | D32 |
| hypocenter | C32⋊C9 | D4 |
| LCS | G ▷ C2×C32⋊C9 ▷ C32⋊C9 | |
| UCS | C1 ◁ C2 | |
110 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 23 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|d^3=f,b^a=bc,e^a=e^2,g^a=g^2,d^b=d^2f^2,f^b=f^2,g^b=g^2,e^d=eg39 conjugacy classes — view the full character table.
| maximal subgroup of | #13047083, #13047088, #13047091, … |
| maximal normal subgroup of | #13047083, #13047088, #13047091, … |
| maximal quotient of | #13046378, #13046382, #13046384, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 55);