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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1440.5223', 'ambient_counter': 5223, 'ambient_order': 1440, 'ambient_tex': 'D_{20}:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 24, 'counter': 141, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1440.5223.30.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '30.f1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 30, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '48.41', 'subgroup_hash': 41, 'subgroup_order': 48, 'subgroup_tex': 'D_{12}:C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1440.5223', 'aut_centralizer_order': 16, 'aut_label': '30.f1', 'aut_quo_index': None, 'aut_stab_index': 30, 'aut_weyl_group': '96.209', 'aut_weyl_index': 480, 'centralizer': '360.f1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['6.f1', '10.d1', '15.a1'], 'contains': ['60.a1', '60.g1', '60.w1', '60.x1', '60.z1', '90.c1'], 'core': '60.a1', 'coset_action_label': None, 'count': 30, 'diagramx': [1435, -1, 4019, -1], 'generators': [13, 8, 2, 1080, 720], 'label': '1440.5223.30.f1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '2.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '15.a1', 'old_label': '30.f1', 'projective_image': '720.813', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '30.f1', 'subgroup_fusion': None, 'weyl_group': '24.14'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 3, 3, 2, 2], 'aut_gens': [[1, 2, 4], [1, 2, 20], [39, 36, 18], [25, 2, 4], [13, 14, 42], [17, 2, 4], [25, 26, 4], [1, 2, 28]], 'aut_group': '288.1028', 'aut_hash': 1028, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 9, 'aut_perms': [7, 5040, 1, 5760, 30, 41040, 90720], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 3, 1], [3, 2, 1, 1], [4, 2, 3, 1], [4, 3, 2, 1], [6, 2, 1, 1], [12, 4, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 3], [3, 2, 1], [4, 2, 3], [4, 3, 2], [6, 2, 1], [12, 4, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 41, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 3], [3, 2, 1, 1], [4, 2, 1, 3], [4, 3, 2, 1], [6, 2, 1, 1], [12, 4, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 112, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '24.14', 'hash': 41, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 6], 'inner_gens': [[1, 2, 44], [1, 2, 28], [9, 26, 4]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 6], [4, 1]], 'label': '48.41', 'linC_count': 9, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D12:C2', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 15, 'number_divisions': 14, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 40, 'number_subgroups': 80, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 19], [3, 2], [4, 12], [6, 2], [12, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [12, 0], 'outer_gens': [[13, 14, 4], [39, 36, 6]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 2]], 'representations': {'PC': {'code': 25788855389011393, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -3, 126, 662, 217, 42, 803, 58, 804]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [91793119281147284, 125101737214347233, 58415889162259849]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [130898247849, 122109387504, 56786428443, 149732242502, 19007034658]}, 'Perm': {'d': 11, 'gens': [3760585, 8852640, 11017584, 15770400, 3]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{12}:C_2', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [4, 4, 12, 2, 12, 12, 12], 'aut_gens': [[1, 2, 4, 24], [289, 132, 1330, 944], [1081, 1332, 138, 1232], [1225, 10, 1220, 312], [1, 1332, 1330, 952], [1, 10, 964, 1128], [1153, 722, 260, 888], [145, 730, 244, 888]], 'aut_group': None, 'aut_hash': 366095071243555349, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 46080, 'aut_permdeg': 32, 'aut_perms': [82720584640097672604460941838405364, 146465228613404099959737815939468407, 209304069472432136146393636261416447, 4943491570227612583744263496044404, 2413460430259888669313948773183574, 98733409461433853018997972689701030, 138762211269932357589504721394635276], 'aut_phi_ratio': 120.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 2, 1], [2, 9, 2, 1], [2, 10, 2, 1], [2, 90, 2, 1], [3, 2, 2, 1], [3, 4, 1, 1], [4, 2, 1, 1], [4, 3, 4, 1], [4, 18, 1, 1], [4, 30, 4, 1], [5, 2, 2, 1], [6, 2, 2, 1], [6, 4, 1, 1], [6, 12, 2, 1], [6, 20, 4, 1], [6, 40, 2, 1], [10, 2, 2, 1], [10, 6, 8, 1], [10, 18, 4, 1], [12, 4, 2, 1], [12, 6, 4, 1], [12, 8, 1, 1], [12, 60, 4, 1], [15, 4, 4, 1], [15, 8, 2, 1], [20, 2, 4, 1], [20, 6, 8, 1], [20, 18, 4, 1], [30, 4, 4, 1], [30, 8, 2, 1], [30, 12, 8, 1], [60, 4, 8, 1], [60, 8, 4, 1], [60, 12, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6^2.C_2^4\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '2880.gq', 'autcentquo_hash': 2229874692249881496, 'autcentquo_nilpotent': False, 'autcentquo_order': 2880, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_3^2:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 2], [2, 9, 2], [2, 10, 2], [2, 90, 2], [3, 2, 2], [3, 4, 1], [4, 2, 1], [4, 3, 4], [4, 18, 1], [4, 30, 4], [5, 2, 2], [6, 2, 2], [6, 4, 1], [6, 12, 2], [6, 20, 4], [6, 40, 2], [10, 2, 2], [10, 6, 8], [10, 18, 4], [12, 4, 2], [12, 6, 4], [12, 8, 1], [12, 60, 4], [15, 4, 4], [15, 8, 2], [20, 2, 4], [20, 6, 8], [20, 18, 4], [30, 4, 4], [30, 8, 2], [30, 12, 8], [60, 4, 8], [60, 8, 4], [60, 12, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '720.813', 'commutator_count': 1, 'commutator_label': '90.10', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 5223, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 2], [2, 9, 1, 2], [2, 10, 1, 2], [2, 90, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [4, 2, 1, 1], [4, 3, 2, 2], [4, 18, 1, 1], [4, 30, 1, 4], [5, 2, 2, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 12, 1, 2], [6, 20, 1, 4], [6, 40, 1, 2], [10, 2, 2, 1], [10, 6, 4, 2], [10, 18, 2, 2], [12, 4, 1, 2], [12, 6, 2, 2], [12, 8, 1, 1], [12, 60, 1, 4], [15, 4, 2, 2], [15, 8, 2, 1], [20, 2, 4, 1], [20, 6, 4, 2], [20, 18, 4, 1], [30, 4, 2, 2], [30, 8, 2, 1], [30, 12, 4, 2], [60, 4, 4, 2], [60, 8, 4, 1], [60, 12, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 26404560, 'exponent': 60, 'exponents_of_order': [5, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 1, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '720.813', 'hash': 5223, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 2, 6, 30], 'inner_gens': [[1, 2, 724, 456], [1, 2, 20, 24], [721, 10, 4, 984], [1009, 2, 484, 24]], 'inner_hash': 813, 'inner_nilpotent': False, 'inner_order': 720, 'inner_split': True, 'inner_tex': 'D_{10}\\times S_3^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 52], [4, 40], [8, 9]], 'label': '1440.5223', 'linC_count': None, 'linC_degree': None, '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': False, 'monomial': True, 'name': 'D20:S3^2', 'ngens': 8, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 36, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 117, 'number_divisions': 64, 'number_normal_subgroups': 144, 'number_subgroup_autclasses': 308, 'number_subgroup_classes': 710, 'number_subgroups': 6668, 'old_label': None, 'order': 1440, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 231], [3, 8], [4, 152], [5, 4], [6, 192], [10, 124], [12, 280], [15, 32], [20, 128], [30, 128], [60, 160]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 4], 'outer_gen_pows': [360, 0, 0, 0], 'outer_gens': [[361, 722, 740, 24], [1, 2, 4, 264], [1, 2, 4, 312], [1, 1092, 602, 512]], 'outer_group': '64.196', 'outer_hash': 196, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [368047, 121, 811440, 142], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2:C_2^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 14], [8, 11], [16, 4], [32, 3]], 'representations': {'PC': {'code': 2972414413502040816153697837367385245235255417221447433978119, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -3, -2, -2, -3, -5, 5769, 17378, 250, 66, 267, 18244, 9860, 116, 43781, 6357, 141, 21510, 14806, 222, 73735]}, 'Perm': {'d': 19, 'gens': [6491569552035120, 14667143952384136, 21345436470758543, 25719601004697600, 33500089616716800, 325, 435, 4359600]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{20}:S_3^2', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
