Linear group L2(8)
Derived Ree group R(3)'


Order = 504 = 23.32.7.
Mult = 1.
Out = 3.

Robert Wilson's ATLAS page for L2(8) is available here.

L2(8): Length 47, 2-generator, 4-relator.

< x, y | x2 = y3 = (xy)7 = (xyxyxy-1xyxyxy-1xy-1)2 = 1 >

Remark: a and b are R.A.Wilson's standard generators for L2(8). We have that [a, b] has order 9.

L2(8): Length 57, 3-generator, 7-relator.

< a, b, c | a2 = b2 = c2 = (ab)3 = (ac)2 = (bc)7 = (abc)9 = 1 >

Remark: This is the Coxeter group G3,7,9.


L2(8):3: Length 83, 2-generator, 5-relator.

< x, y | x2 = y3 = (xy)9 = [x, y]9 = (xyxyxy-1xyxy-1xy-1)2 = 1 >

26.L2(8): Length 139, 2-generator, 5-relator.

< x, y | x2 = y3 = (xy)7 = [x, y]9 = (xyxyxy-1xyxy-1xy-1)7 = 1 >

Remark: This is a non-split extension of 26 by L2(8). We may obtain a faithful permutation representation of G = 26.L2(8) of degree 72 over H = < yx, y-1xyxy-1xyxyxy-1xy-1xyxy-1xyxy >.

Z7.L2(8): Length 127, 2-generator, 5-relator.

< x, y | x2 = y3 = (xy)7 = [x, y]9 = (xyxyxy-1xy-1)9 = 1 >

Remark: This is a non-split extension of Z7 by L2(8).

Realisation:
x =
1 0 0 0 0 0 0 0
0 -1 0 0 0 0 0 0
0 0 -1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 1 0 0 0 0
0 -1 0 0 0 0 1 -1
0 0 1 0 0 1 0 1
0 1 1 0 0 0 0 1
y =
1 0 0 0 0 0 0 0
-1 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
1 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
1 1 1 1 -1 -1 0 0
1 -1 -1 0 0 0 0 -1
0 -1 0 0 0 0 1 -1

(L2(8) x 22):3: Length 47, 2-generator, 4-relator.

< x, y | x2 = y3 = (xy)9 = (xyxyxy-1xyxy-1xy-1)2 = 1 >
- -

- Last updated 26th June, 1997
- John N. Bray