Finite Group DatabasebetaFinite Group Databasebeta| order | 12348 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C7 · Schur multiplier (H2) = C2×C3 |
| direct product | A4×(C49⋊C3)⋊C7 |
| Fitting, Hall | C22×7−1+2 ⋊ C32 |
| derived | C22×C49 ⋊ C32×C7 |
| Sylow | 7−1+2 ⋊ C3×A4 |
| Sylow | C22 ⋊ C3×(C49⋊C3)⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C49 | C32×C7 |
| Frattini | C7 | C7×A4×ΓL(1,𝔽8) |
| Fitting | C22×7−1+2 | C32 |
| socle | C22×C7 | C3×C7×ΓL(1,𝔽8) |
| LCS | G ▷ C22×C49 | |
| UCS | C1 | |
94 conjugacy classes of subgroups.
| transitive degree | μt(G) = 196 |
148 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#72215772, S4×(C49⋊C3)⋊C7, #72275559 |
| maximal normal subgroup of | A4×#72215772, S4×(C49⋊C3)⋊C7, #72275559 |
Past the SmallGroups library coverage — no SmallGroups id.