-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1660.4', 'ambient_counter': 4, 'ambient_order': 1660, 'ambient_tex': 'C_{1660}', 'central': True, 'central_factor': False, 'centralizer_order': 1660, 'characteristic': True, 'core_order': 415, 'counter': 3, 'cyclic': True, 'direct': True, 'hall': 415, 'label': '1660.4.4.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': ['C2'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '415.1', 'subgroup_hash': 1, 'subgroup_order': 415, 'subgroup_tex': 'C_{415}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1660.4', 'aut_centralizer_order': 2, 'aut_label': '4.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '328.9', 'aut_weyl_index': 2, 'centralizer': '1.a1.a1', 'complements': ['415.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['20.a1.a1', '332.a1.a1'], 'core': '4.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1818, 1818, 1818, 1818, 1147, 1147, 1147, 1147], 'generators': [664, 20], 'label': '1660.4.4.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '4.a1.a1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['20.a1.a1', '332.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.a1.a1', 'projective_image': '4.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '4.1', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '4.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '415.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 164, 'aut_gen_orders': [2, 164], 'aut_gens': [[1], [331], [207]], 'aut_group': '328.9', 'aut_hash': 9, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 328, 'aut_permdeg': 86, 'aut_perms': [240653664262646398824364291566343783056841073003064150375214511720418715778479274250420219239686814127869506093056000000000, 285064341596380282813238570119582695616465865745660975498642690668824353012129487853579065019958376597791165375861227520000000000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [83, 1, 82, 1], [415, 1, 328, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{164}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 164, 'autcent_group': '328.9', 'autcent_hash': 9, 'autcent_nilpotent': True, 'autcent_order': 328, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{164}', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [5, 1, 4], [83, 1, 82], [415, 1, 328]], 'center_label': '415.1', 'center_order': 415, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['5.1', '83.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['83.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [83, 1, 82, 1], [415, 1, 328, 1]], 'element_repr_type': 'PC', 'elementary': 415, 'eulerian_function': 1, 'exponent': 415, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 41], 'factors_of_order': [5, 83], 'faithful_reps': [[1, 0, 328]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '415.1', 'hash': 1, 'hyperelementary': 415, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 328, 'irrQ_dim': 328, 'irrR_degree': 2, 'irrep_stats': [[1, 415]], 'label': '415.1', 'linC_count': 328, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 86, 'linQ_degree_count': 1, 'linQ_dim': 86, 'linQ_dim_count': 1, 'linR_count': 164, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C415', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 415, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 415, 'order_factorization_type': 11, 'order_stats': [[1, 1], [5, 4], [83, 82], [415, 328]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 164, 'outer_gen_orders': [2, 164], 'outer_gen_pows': [0, 0], 'outer_gens': [[331], [207]], 'outer_group': '328.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 328, 'outer_permdeg': 86, 'outer_perms': [240653664262646398824364291566343783056841073003064150375214511720418715778479274250420219239686814127869506093056000000000, 285064341596380282813238570119582695616465865745660975498642690668824353012129487853579065019958376597791165375861227520000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{164}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 88, 'pgroup': 0, 'primary_abelian_invariants': [5, 83], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [82, 1], [328, 1]], 'representations': {'PC': {'code': 33947, 'gens': [1], 'pres': [2, -5, -83, 10]}, 'GLFp': {'d': 2, 'p': 331, 'gens': [3696897064]}, 'Perm': {'d': 88, 'gens': [8431029193518110868854402074797558380919134952245424849291890044858616462910372696322733692945659174018937887129600000000000000000000, 38979875363505302337055332973131205624792690113575452946607196594293007568261730186266584808295291703787520000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [415], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{415}', 'transitive_degree': 415, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1660.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 164, 'aut_gen_orders': [2, 2, 164], 'aut_gens': [[1], [1161], [831], [1037]], 'aut_group': '656.46', 'aut_hash': 46, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 656, 'aut_permdeg': 88, 'aut_perms': [240653664262646398824364291566343783056841073003064150375214511720418715778479274250420219239686814127869506093056000000000, 2107757298379527717213600518699389595229783738061356212322972511214654115727593174080683423236414793504734471782400000000000000000000, 285064341596380282813238570119582695616465865745660975498642690668824353012129487853579065019958376597791165375861227520000000000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1], [83, 1, 82, 1], [166, 1, 82, 1], [332, 1, 164, 1], [415, 1, 328, 1], [830, 1, 328, 1], [1660, 1, 656, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_{164}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 164, 'autcent_group': '656.46', 'autcent_hash': 46, 'autcent_nilpotent': True, 'autcent_order': 656, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{164}', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2], [5, 1, 4], [10, 1, 4], [20, 1, 8], [83, 1, 82], [166, 1, 82], [332, 1, 164], [415, 1, 328], [830, 1, 328], [1660, 1, 656]], 'center_label': '1660.4', 'center_order': 1660, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '5.1', '83.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1], ['83.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1], [83, 1, 82, 1], [166, 1, 82, 1], [332, 1, 164, 1], [415, 1, 328, 1], [830, 1, 328, 1], [1660, 1, 656, 1]], 'element_repr_type': 'PC', 'elementary': 830, 'eulerian_function': 1, 'exponent': 1660, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 41], 'factors_of_order': [2, 5, 83], 'faithful_reps': [[1, 0, 656]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '830.4', 'hash': 4, 'hyperelementary': 830, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 656, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 1660]], 'label': '1660.4', 'linC_count': None, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1660', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 1660, 'number_divisions': 12, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 12, 'old_label': None, 'order': 1660, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 4], [10, 4], [20, 8], [83, 82], [166, 82], [332, 164], [415, 328], [830, 328], [1660, 656]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 164, 'outer_gen_orders': [2, 2, 164], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1161], [831], [1037]], 'outer_group': '656.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 656, 'outer_permdeg': 88, 'outer_perms': [240653664262646398824364291566343783056841073003064150375214511720418715778479274250420219239686814127869506093056000000000, 2107757298379527717213600518699389595229783738061356212322972511214654115727593174080683423236414793504734471782400000000000000000000, 285064341596380282813238570119582695616465865745660975498642690668824353012129487853579065019958376597791165375861227520000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{164}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 92, 'pgroup': 0, 'primary_abelian_invariants': [4, 5, 83], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1], [82, 2], [164, 1], [328, 2], [656, 1]], 'representations': {'PC': {'code': 66987401966620599, 'gens': [1], 'pres': [4, -2, -2, -5, -83, 8, 21, 70]}, 'GLFp': {'d': 2, 'p': 331, 'gens': [11925554974]}, 'Perm': {'d': 92, 'gens': [408571890232484411760118751745727077918531739353289410909109637518870983100868116050147836126993116402544739479243980800000000000000000000000, 135216660723001204716247785678757595766896706143143174333596648117861503942594373275424683867321237028413897417151007948800000000000000000000, 8431029193518110868854402074797558380919134952245424849291890044858616462910372696322733692945659174018937887129600000000000000000000, 38979875363505302337055332973131205624792690113575452946607196594293007568261730186266584808295291703787520000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [1660], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{1660}', 'transitive_degree': 1660, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
