Linear group L3(2)
Linear group L2(7)

Order = 168 = 23.3.7.
Mult = 2.
Out = 2.
Robert Wilson's ATLAS page for L3(2) is available here.

L3(2): Length 35, 2-generator, 4-relator.

< x, y | x2 = y3 = (xy)7 = [x, y]4 = 1 >

Remark: This is the Coxeter group (2, 3, 7; 4).
Reciprocity: (x, y) and (x, y-1) are conjugate in Aut(L2(7)) = PGL2(7) but not in L2(7).

PGL2(7): Length 37, 2-generator, 4-relator.

< x, y | x2 = y3 = (xy)8 = [x, y]4 = 1 >

Remark: This is the Coxeter group (2, 3, 8; 4).
Reciprocity: (x, y) and (x, y-1) are conjugate in PGL2(7) = G and also in L2(7) = G'.

PGL2(7): Length 28, 2-generator, 4-relator.

< x, y | x8 = (xy)4 = (y-1x)3 = (xy-1x)2 = 1 >

Remark: The relations y8 = xy-1xyx-1y = 1 hold.

23.L3(2): Length 55, 2-generator, 4-relator.

< x, y | x2 = y3 = (xy)7 = (xyxyxy-1xyxy-1xy-1)3 = 1 >

Remark: G is a non-split extension of 23 by L3(2). The relation [x, y]8 = 1 holds.
Reciprocity: (x, y) and (x, y-1) are not conjugate in Aut(G).

(L2(7) x 4).2: Length 22, 2-generator, 3-relator.

< x, y | x8 = (xy)4 = xy-1xyx-1y = 1 >

Remark: The relation y8 = 1 holds. We have that G has PGL2(7) as a homomorphic image and that G/G' is cyclic of order 8.

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- Last updated 16th July, 1997
- John N. Bray