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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1144.15', 'ambient_counter': 15, 'ambient_order': 1144, 'ambient_tex': 'D_{13}\\times C_{44}', 'central': False, 'central_factor': False, 'centralizer_order': 572, 'characteristic': True, 'core_order': 13, 'counter': 23, 'cyclic': True, 'direct': False, 'hall': 13, 'label': '1144.15.88.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '88.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '88.8', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 8, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 88, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_{44}', 'simple': True, 'solvable': True, 'special_labels': ['D', 'L1', 'D1', 'C4'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '13.1', 'subgroup_hash': 1, 'subgroup_order': 13, 'subgroup_tex': 'C_{13}', 'supersolvable': True, 'sylow': 13}
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gps_subgroup_data • Show schema
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{'ambient': '1144.15', 'aut_centralizer_order': 520, 'aut_label': '88.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '12.2', 'aut_weyl_index': 520, 'centralizer': '2.b1.a1', 'complements': ['13.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '44.a1.a1', '44.b1.a1', '44.b1.b1'], 'contains': ['1144.a1.a1'], 'core': '88.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1100, 2824, 344, 3626, 3976, 5465, 3174, 4997], 'generators': [88], 'label': '1144.15.88.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '88.a1.a1', 'normal_contained_in': ['8.a1.a1', '44.a1.a1', '44.b1.a1', '44.b1.b1'], 'normal_contains': ['1144.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '88.a1.a1', 'projective_image': '1144.15', 'quotient_action_image': '2.1', 'quotient_action_kernel': '44.2', 'quotient_action_kernel_order': 44, 'quotient_fusion': None, 'short_label': '88.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '13.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 12, 'aut_gen_orders': [12], 'aut_gens': [[1], [2]], 'aut_group': '12.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 12, 'aut_permdeg': 7, 'aut_perms': [867], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [13, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{12}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 12, 'autcent_group': '12.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{12}', '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], [13, 1, 12]], 'center_label': '13.1', 'center_order': 13, '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': ['13.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [13, 1, 12, 1]], 'element_repr_type': 'PC', 'elementary': 13, 'eulerian_function': 1, 'exponent': 13, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [13], 'faithful_reps': [[1, 0, 12]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '13.1', 'hash': 1, 'hyperelementary': 13, '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': 12, 'irrQ_dim': 12, 'irrR_degree': 2, 'irrep_stats': [[1, 13]], 'label': '13.1', 'linC_count': 12, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 1, 'linQ_dim': 12, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C13', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 13, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 13, 'order_factorization_type': 1, 'order_stats': [[1, 1], [13, 12]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [12], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '12.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [867], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 13, 'pgroup': 13, 'primary_abelian_invariants': [13], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [12, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -13]}, 'Lie': [{'d': 1, 'q': 13, 'gens': [522956313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 13, 'gens': [2211]}, 'Perm': {'d': 13, 'gens': [5748019200]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [13], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{13}', 'transitive_degree': 13, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '88.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 780, 'aut_gen_orders': [2, 2, 10, 12, 13], 'aut_gens': [[1, 2], [573, 882], [1, 310], [1, 466], [1, 90], [89, 2]], 'aut_group': None, 'aut_hash': 1837704963680868734, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6240, 'aut_permdeg': 38, 'aut_perms': [197924156413080889705658550818062584911584286, 440832090912137663351943530771820252733299156, 440945763179064625911098894955755329197795643, 55441325016218370858235813865408134134259872, 84836674162949293542213638987576300674834872], 'aut_phi_ratio': 13.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 13, 2, 1], [4, 1, 2, 1], [4, 13, 2, 1], [11, 1, 10, 1], [13, 2, 6, 1], [22, 1, 10, 1], [22, 13, 20, 1], [26, 2, 6, 1], [44, 1, 20, 1], [44, 13, 20, 1], [52, 2, 12, 1], [143, 2, 60, 1], [286, 2, 60, 1], [572, 2, 120, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{65}.(C_2^3\\times C_{12})', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 10, 'autcent_group': '40.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 40, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 156, 'autcentquo_group': '156.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 156, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{13}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 13, 2], [4, 1, 2], [4, 13, 2], [11, 1, 10], [13, 2, 6], [22, 1, 10], [22, 13, 20], [26, 2, 6], [44, 1, 20], [44, 13, 20], [52, 2, 12], [143, 2, 60], [286, 2, 60], [572, 2, 120]], 'center_label': '44.2', 'center_order': 44, 'central_product': True, 'central_quotient': '26.1', 'commutator_count': 1, 'commutator_label': '13.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '11.1', '13.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['26.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 13, 1, 2], [4, 1, 2, 1], [4, 13, 2, 1], [11, 1, 10, 1], [13, 2, 6, 1], [22, 1, 10, 1], [22, 13, 10, 2], [26, 2, 6, 1], [44, 1, 20, 1], [44, 13, 20, 1], [52, 2, 12, 1], [143, 2, 60, 1], [286, 2, 60, 1], [572, 2, 120, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 572, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 13], 'factors_of_order': [2, 11, 13], 'faithful_reps': [[2, 0, 120]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '572.8', 'hash': 15, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 26, 'inner_gen_orders': [2, 13], 'inner_gens': [[1, 882], [265, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 26, 'inner_split': False, 'inner_tex': 'D_{13}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 240, 'irrQ_dim': 240, 'irrR_degree': 4, 'irrep_stats': [[1, 88], [2, 264]], 'label': '1144.15', 'linC_count': 120, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 16, 'linQ_dim': 24, 'linQ_dim_count': 16, 'linR_count': 300, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D13*C44', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 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': 16, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 352, 'number_divisions': 18, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 32, 'number_subgroups': 152, 'old_label': None, 'order': 1144, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 27], [4, 28], [11, 10], [13, 12], [22, 270], [26, 12], [44, 280], [52, 24], [143, 120], [286, 120], [572, 240]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 2, 2, 30], 'outer_gen_pows': [0, 0, 0, 1], 'outer_gens': [[573, 882], [1, 310], [1, 570], [1, 194]], 'outer_group': '240.208', 'outer_hash': 208, 'outer_nilpotent': True, 'outer_order': 240, 'outer_permdeg': 16, 'outer_perms': [1307674368000, 6227020800, 40279680, 368673], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{30}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [10, 4], [12, 2], [20, 2], [24, 1], [120, 2], [240, 1]], 'representations': {'PC': {'code': 100533250147445830481319585911, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -11, -13, 8821, 26, 9302, 42, 1923, 238, 26404]}, 'Perm': {'d': 28, 'gens': [21010008464047, 44548106803200, 11308319599659380338360320000, 21009968179200, 559571293]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 44], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{13}\\times C_{44}', 'transitive_degree': 572, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '88.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [2, 2, 20], 'aut_gens': [[1, 2], [1, 42], [45, 2], [45, 7]], 'aut_group': '80.46', 'aut_hash': 46, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 80, 'aut_permdeg': 11, 'aut_perms': [720, 766080, 10887184], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1], [11, 1, 10, 1], [22, 1, 10, 1], [22, 1, 20, 1], [44, 1, 40, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times C_{10}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '80.46', 'autcent_hash': 46, 'autcent_nilpotent': True, 'autcent_order': 80, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4\\times C_{10}', '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, 3], [4, 1, 4], [11, 1, 10], [22, 1, 30], [44, 1, 40]], 'center_label': '88.8', 'center_order': 88, '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': ['2.1', '2.1', '2.1', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [11, 1, 10, 1], [22, 1, 10, 3], [44, 1, 20, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 36, 'exponent': 44, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '44.4', 'hash': 8, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 88]], 'label': '88.8', 'linC_count': 1440, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 4, 'linR_count': 40, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C44', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 88, 'number_divisions': 12, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 88, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 4], [11, 10], [22, 30], [44, 40]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [2, 2, 20], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 42], [45, 2], [45, 7]], 'outer_group': '80.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 80, 'outer_permdeg': 11, 'outer_perms': [720, 766080, 10887184], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_{10}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [10, 4], [20, 2]], 'representations': {'PC': {'code': 506940009, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -11, 21, 34]}, 'GLFp': {'d': 2, 'p': 89, 'gens': [23968947, 7754740]}, 'Perm': {'d': 17, 'gens': [273988915200, 20922789888000, 36288000, 87657292800]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 44], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{44}', 'transitive_degree': 88, 'wreath_data': None, 'wreath_product': False}
