Triple

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Phrase: symmetric group S_4 | Statement: [Fermat surface, hasAutomorphismGroupContaining, symmetric group S_4]

Target entity: symmetric group S_4
Context triple: [Fermat surface, hasAutomorphismGroupContaining, symmetric group S_4]

• A. symmetric group S5
  The symmetric group S5 is the group of all permutations of five elements, a fundamental finite group of order 120 that plays a key role in group theory and Galois theory.
• B. Coxeter group
  A Coxeter group is an abstract group generated by reflections across hyperplanes, fundamental in the classification and study of regular polytopes, tessellations, and symmetries in geometry and algebra.
• C. SL(2,7)
  SL(2,7) is the special linear group of 2×2 matrices with determinant 1 over the finite field with 7 elements, a non-abelian finite group of order 336 that plays an important role in group theory and geometry.
• D. Schreier
  Schreier is a surname of Germanic origin borne by various notable individuals, including mathematicians and other professionals.
• E. Burnside's lemma
  Burnside's lemma is a result in group theory and combinatorics that counts distinct configurations under symmetries by averaging the number of fixed points of group actions.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: symmetric group S_4
Target entity description: The symmetric group S₄ is the group of all 24 permutations of four elements, a fundamental example in group theory and a key object in the study of symmetries in algebra and geometry.

• A. symmetric group S5
  The symmetric group S5 is the group of all permutations of five elements, a fundamental finite group of order 120 that plays a key role in group theory and Galois theory.
• B. Coxeter group
  A Coxeter group is an abstract group generated by reflections across hyperplanes, fundamental in the classification and study of regular polytopes, tessellations, and symmetries in geometry and algebra.
• C. SL(2,7)
  SL(2,7) is the special linear group of 2×2 matrices with determinant 1 over the finite field with 7 elements, a non-abelian finite group of order 336 that plays an important role in group theory and geometry.
• D. Schreier
  Schreier is a surname of Germanic origin borne by various notable individuals, including mathematicians and other professionals.
• E. Burnside's lemma
  Burnside's lemma is a result in group theory and combinatorics that counts distinct configurations under symmetries by averaging the number of fixed points of group actions.
• F. None of above. chosen