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In the case that two of the three values <matH>a,b,c</math> are equal and the third is different (thus, we may have an [[oblate spheroid]] or a [[prolate spheroid]]) the isometry group is infinite and can be described as the direct product of the generalized dihedral group corresponding to the circle group and the cyclic group of order two. The orientation-preserving isometry group is simply the generalized dihedral group corresponding to the circle group.

In the case that all three values <math>a,b,c</math> are equal, we get a [[2-sphere in Euclidean space]]. The isometry group is the orthogonal group <math>O(3,\R)</math> and the orientation-preserving isometry group is the ''special'' orthogonal group <math>SO(3,\R)</math>.

==Verification of theorems==

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