Linear group L₃(4)
Mathieu group M₂₁

Order = 20 160 = 2⁶.3².5.7.
Mult = 3 x 4².
Out = D₁₂.

Robert Wilson's ATLAS page for L₃(4) is available here.

< x, y | x² = y⁴ = (xy)⁷ = (xy²)⁵ = (xyxyxy^-1)⁵ = xyxyxy²xyxy^-1xy²xyxy²xy^-1xy²xyxyxy^-1xy^-1xy² = 1 >

Remark: x and y are R.A.Wilson's standard generators for L₃(4).

< x, y | x² = y⁴ = (xy)⁷ = (xy²)⁵ = (xyxyxy^-1)⁵ = (xyxyxy²xy^-1)⁵ = 1 >

Remark: On the same generators as the previous presentation.

< x, y | x² = y⁴ = (xy)⁷ = (xy²)⁵ = (xyxyxy^-1)⁵ = 1 >

Remark: Mapping on to standard generators for L₃(4). A central element of order 4 is (xyxyxy²xy^-1)⁵.

Some subgroups:

H1 = < x(xyxyxy²xy^-1)¹⁰, yxy^-1xyxy² > = A₆ of order 360 and index 224 = 4 × 56.
H2 = < x, yxy^-1xy^-1xy > = L₃(2) of order 168 and index 480 = 4 × 120.

It is very difficult to enumerate the cosets of H1 in G using MAGMA.

Last updated 25^th August, 1997

John N. Bray

Linear group L3(4) Mathieu group M21