Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #4175 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C4×#4175 |
| Fitting | C4×AΣL(1,𝔽27) ⋊ C2 |
| derived | C33 ⋊ C2×C3×C4 |
| Sylow | AΣL(1,𝔽27) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4175 |
| commutator, hypocenter | C33 | C2×C3×C4 |
| Frattini | C2×C32 | C2×C3×D3 |
| Fitting | C4×AΣL(1,𝔽27) | C2 |
| socle | C2×C3 | C2×(C3⋊D3)⋊C3 |
| prime radical | C2×#4175 | C2 |
| cosocle | C2×C33 | C22×C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C4 | |
134 conjugacy classes of subgroups.
| transitive degree | μt(G) = 36 |
52 conjugacy classes — view the full character table.
| maximal subgroup of | #4175⋊C8, #13046412, C8×#4175, … |
| maximal normal subgroup of | #4175⋊C8, #13046412, C8×#4175, … |
| maximal quotient of | C8×#4175, #13046562, C4×#66322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 130);