Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C22×C3×C27 ⋊ C2 |
| Sylow | C3×C27 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×C3⋊D27 |
| commutator, cosocle | C2×C3×C27 | C22 |
| Frattini | C2×C9 | C2×C3⋊D3 |
| Fitting | C22×C3×C27 | C2 |
| socle | C2×C32 | C2×D9 |
| hypercenter | C22 | C3⋊D27 |
| hypocenter | C3×C27 | D4 |
| LCS | G ▷ C2×C3×C27 ▷ C3×C27 | |
| UCS | C1 ◁ C2 ◁ C22 | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
2^3,3^4|d^3=f,f^3=g,b^a=bc,d^a=d^2f^2g^2,e^a=e^2,f^a=f^2g^2,g^a=g^2165 conjugacy classes — view the full character table.
| maximal subgroup of | #13047176, #13047181, #67×D27, … |
| maximal normal subgroup of | #13047176, #13047181, #67×D27, … |
| maximal quotient of | #13046778, #13046780, #13046809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 196);