Let
Α be the center of
α with the axes of the glide-reflection
γ passing through
Α. Since (
αγ)
2 is a minimal translation (with the conventions of qn 35(ii)), we consider a fundamental square of which
Α,
Αγ2 and
Α (
αγ)
2 are three of the vertices. The transformation
α2γ and its conjugates by
α,
γ and
γα give four reflections with axes parallel to the diagonals of the fundamental square. Six glide-reflections are obtained from
γ by repeated conjugation by
α and
γ with axes parallel to the diagonals of the fundamental square and intersecting the square. These axes all contain 4-centres. The transformation
αγ is a glide-reflection with axis parallel to the translation (
αγ)
2 and by successive conjugation by the quarter-turns
α–1, (
αγ2)
–1, and
γ–2α–1γ2 we obtain four more glide-reflections with axes parallel to the sides of the fundamental square, but not passing through any 4-centre.
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