Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C81 ⋊ C2 |
| Sylow | C81 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D81 |
| commutator, cosocle | C2×C81 | C22 |
| Frattini | C2×C27 | C2×D3 |
| Fitting | C22×C81 | C2 |
| socle | C2×C3 | C2×D27 |
| hypercenter | C22 | D81 |
| hypocenter | C81 | D4 |
| LCS | G ▷ C2×C81 ▷ C81 | |
| UCS | C1 ◁ C2 ◁ C22 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
2^3,3^4|d^3=e,e^3=f,f^3=g,b^a=bc,d^a=d^2e^2f^2g^2,e^a=e^2f^2g^2,f^a=f^2g^2,g^a=g^2165 conjugacy classes — view the full character table.
| maximal subgroup of | #13046348, D4×D81, #13046350, … |
| maximal normal subgroup of | #13046348, D4×D81, #13046350, … |
| maximal quotient of | Dic81⋊C4, (C2×D81)⋊C4, #13046297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 8);