Finite Group DatabasebetaFinite Group Databasebeta| order | 6174 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C7 · Schur multiplier (H2) = C3 |
| direct product | C3×#72215772 |
| Fitting | C3×7−1+2 ⋊ C2×C3 |
| derived | C49 ⋊ C2×C32×C7 |
| Hall | C3×(C49⋊C3)⋊C7 ⋊ C2 |
| Hall | D49⋊C7 ⋊ C32 |
| Sylow | 7−1+2 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #72215772 |
| commutator, cosocle, hypocenter | C49 | C2×C32×C7 |
| Frattini | C7 | C3×C7×AGL(1,𝔽7) |
| Fitting | C3×7−1+2 | C2×C3 |
| socle | C3×C7 | C7×AGL(1,𝔽7) |
| LCS | G ▷ C49 | |
| UCS | C1 ◁ C3 | |
74 conjugacy classes of subgroups.
| transitive degree | μt(G) = 147 |
150 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72215772, A4×#72215772 |
| maximal normal subgroup of | D3×#72215772 |
| maximal quotient of | A4×#72215772 |
Past the SmallGroups library coverage — no SmallGroups id.