Finite
Geometry
Notes |

Reflection Group

by Steven H. Cullinane Sept. 14, 2009

The image below is an illustration for

"A Reflection Group of Order 168"

and "The Eightfold Cube."

The calculation indicates that Klein's

simple group of order 168 is generated

by permuting parallel 1x1x2 "bricks" in

each of three halves of a 2x2x2 cube.

Each such permutation is a*reflection*:

a linear transformation of a vector space

that fixes a hyperplane pointwise.

"A Reflection Group of Order 168"

and "The Eightfold Cube."

The calculation indicates that Klein's

simple group of order 168 is generated

by permuting parallel 1x1x2 "bricks" in

each of three halves of a 2x2x2 cube.

Each such permutation is a

a linear transformation of a vector space

that fixes a hyperplane pointwise.

Here "A(1, 7)" is a MAGMA designation

for the group L

which is isomorphic to the group L

The group action illustrated by the permutations

in the calculation is of course that of L

The group L

only three of the six permutations above--