Finite Group DatabasebetaFinite Group Databasebeta| order | 12054 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×ΓL(1,𝔽8)×D41 |
| unambiguous semidirect product | C41⋊(C2×C7×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C7×D41)⋊(C3×C7) |
| Fitting, socle, Hall | C72×C41 ⋊ C2×C3 |
| derived | C7×C41 ⋊ C2×C3×C7 |
| Hall | C7×ΓL(1,𝔽8)×C41 ⋊ C2 |
| Hall | C72×D41 ⋊ C3 |
| Sylow | C72 ⋊ C3×D41 |
| Sylow | C41 ⋊ C2×C7×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | ΓL(1,𝔽8)×D41 |
| commutator, cosocle, hypocenter | C7×C41 | C2×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C72×C41 | C2×C3 |
| LCS | G ▷ C7×C41 | |
| UCS | C1 ◁ C7 | |
40 conjugacy classes of subgroups.
770 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12054, 16);