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