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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1760.415', 'ambient_counter': 415, 'ambient_order': 1760, 'ambient_tex': 'C_{11}:C_{10}\\times Q_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 44, 'characteristic': True, 'core_order': 352, 'counter': 20, 'cyclic': False, 'direct': False, 'hall': 22, 'label': '1760.415.5.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '5.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '5.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 5, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5', 'simple': False, 'solvable': True, 'special_labels': ['F'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '352.169', 'subgroup_hash': 169, 'subgroup_order': 352, 'subgroup_tex': 'Q_{16}\\times C_{22}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.415', 'aut_centralizer_order': 11, 'aut_label': '5.a1', 'aut_quo_index': 4, 'aut_stab_index': 1, 'aut_weyl_group': None, 'aut_weyl_index': 11, 'centralizer': '40.a1.a1', 'complements': ['352.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['10.a1.a1', '10.a1.b1', '10.b1.a1', '10.c1.a1', '10.c1.b1', '10.c1.c1', '10.c1.d1', '55.a1.a1'], 'core': '5.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [9674, 803, 1219, 8131, 7407, 7742, 6965, 7001], 'generators': [1, 880, 220, 160, 10, 1320], 'label': '1760.415.5.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '5.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['10.a1.b1', '10.a1.a1', '10.b1.a1', '10.c1.b1', '10.c1.d1', '10.c1.a1', '10.c1.c1', '55.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '5.a1.a1', 'projective_image': '440.13', 'quotient_action_image': '5.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '5.a1.a1', 'subgroup_fusion': None, 'weyl_group': '40.10'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '88.12', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 40, 'aut_gen_orders': [10, 20, 20, 40, 20, 10], 'aut_gens': [[1, 2, 4], [1, 267, 332], [1, 266, 52], [177, 266, 116], [1, 46, 292], [177, 3, 157], [1, 178, 29]], 'aut_group': None, 'aut_hash': 8718903400604585133, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 2560, 'aut_permdeg': 26, 'aut_perms': [15942248410501883247080245, 32560526081052592472787870, 41881888934191841293931365, 242487462411925526488502461, 265863504416405920599235513, 47801288400922473621512887], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 1, 2], [4, 4, 4, 1], [8, 2, 4, 1], [11, 1, 10, 1], [22, 1, 10, 1], [22, 1, 20, 1], [44, 2, 10, 2], [44, 4, 40, 1], [88, 2, 40, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{10}\\times C_2^3.C_2^4.C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '320.1523', 'autcent_hash': 1523, 'autcent_nilpotent': True, 'autcent_order': 320, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5:C_{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 4, 4], [8, 2, 4], [11, 1, 10], [22, 1, 30], [44, 2, 20], [44, 4, 40], [88, 2, 40]], 'center_label': '44.4', 'center_order': 44, 'central_product': True, 'central_quotient': '8.3', 'commutator_count': 1, 'commutator_label': '4.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 169, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['16.9', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 4, 1, 4], [8, 2, 2, 2], [11, 1, 10, 1], [22, 1, 10, 3], [44, 2, 10, 2], [44, 4, 10, 4], [88, 2, 20, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 5586, 'exponent': 88, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '88.12', 'hash': 169, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [1, 2, 4], 'inner_gens': [[1, 2, 4], [1, 2, 92], [1, 266, 4]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 88], [2, 66]], 'label': '352.169', 'linC_count': 1920, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 8, 'linQ_dim': 18, 'linQ_dim_count': 8, 'linR_count': 80, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'Q16*C22', 'ngens': 6, 'nilpotency_class': 3, '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': 14, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 154, 'number_divisions': 24, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 60, 'number_subgroups': 76, 'old_label': None, 'order': 352, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 20], [8, 8], [11, 10], [22, 30], [44, 200], [88, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [2, 2, 2, 2, 20], 'outer_gen_pows': [348, 0, 348, 0, 8], 'outer_gens': [[177, 47, 180], [1, 3, 260], [1, 47, 4], [1, 3, 180], [1, 46, 101]], 'outer_group': '320.1523', 'outer_hash': 1523, 'outer_nilpotent': True, 'outer_order': 320, 'outer_permdeg': 15, 'outer_perms': [12936682800, 3719520, 12936762720, 186810714720, 93888074256], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5:C_{10}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 29, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 11], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 2], [10, 8], [20, 2], [40, 2]], 'representations': {'PC': {'code': 91227128359534201886000009, 'gens': [1, 2, 3], 'pres': [6, -2, -2, -2, -2, -2, -11, 1063, 836, 50, 2217, 69, 88]}, 'GLZN': {'d': 2, 'p': 46, 'gens': [2477581, 4380165, 2433425, 98395, 935401, 2485475]}, 'Perm': {'d': 29, 'gens': [340102076874088888860062284800, 654686099378026517004113433600, 479001600, 36288000, 971984673007516468907121715200, 1288268279492728413126094848000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 22], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_{16}\\times C_{22}', 'transitive_degree': 352, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 440, 'aut_gen_orders': [40, 20, 20, 10, 40, 22], 'aut_gens': [[1, 2, 20], [881, 1663, 500], [1, 762, 341], [1, 1463, 700], [1, 243, 700], [881, 1143, 501], [881, 362, 460]], 'aut_group': None, 'aut_hash': 2016333457522889885, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 28160, 'aut_permdeg': 27, 'aut_perms': [10369587246467483896247349409, 4270639055280423088250262837, 6165721406643483447285903212, 8465483255827783683968956205, 2370819651866855753427559784, 4484037795055987019684416173], 'aut_phi_ratio': 44.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 1, 2], [4, 4, 4, 1], [5, 11, 1, 4], [8, 2, 4, 1], [10, 11, 1, 4], [10, 11, 2, 4], [11, 5, 2, 1], [20, 22, 1, 8], [20, 44, 4, 4], [22, 5, 2, 1], [22, 5, 4, 1], [40, 22, 4, 4], [44, 10, 2, 2], [44, 20, 8, 1], [88, 10, 8, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2\\times C_{44}).C_{10}.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 220, 'autcentquo_group': '880.118', 'autcentquo_hash': 118, 'autcentquo_nilpotent': False, 'autcentquo_order': 880, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 4, 4], [5, 11, 4], [8, 2, 4], [10, 11, 12], [11, 5, 2], [20, 22, 8], [20, 44, 16], [22, 5, 6], [40, 22, 16], [44, 10, 4], [44, 20, 8], [88, 10, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '440.13', 'commutator_count': 1, 'commutator_label': '44.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 415, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.9', 1], ['2.1', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 4, 1, 4], [5, 11, 4, 1], [8, 2, 2, 2], [10, 11, 4, 3], [11, 5, 2, 1], [20, 22, 4, 2], [20, 44, 4, 4], [22, 5, 2, 3], [40, 22, 8, 2], [44, 10, 2, 2], [44, 20, 2, 4], [88, 10, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 62496, 'exponent': 440, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '440.44', 'hash': 415, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 220, 'inner_gen_orders': [1, 10, 44], 'inner_gens': [[1, 2, 20], [1, 2, 300], [1, 1482, 20]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 440, 'inner_split': False, 'inner_tex': 'C_{44}:C_{10}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 30], [5, 16], [10, 12]], 'label': '1760.415', 'linC_count': 160, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 8, 'linQ_dim': 18, 'linQ_dim_count': 8, 'linR_count': 48, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C11:C10*Q16', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 42, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 98, 'number_divisions': 36, 'number_normal_subgroups': 66, 'number_subgroup_autclasses': 64, 'number_subgroup_classes': 120, 'number_subgroups': 532, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [4, 20], [5, 44], [8, 8], [10, 132], [11, 10], [20, 880], [22, 30], [40, 352], [44, 200], [88, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [220, 1540, 0, 440, 440], 'outer_gens': [[881, 223, 420], [1, 223, 900], [1, 882, 1300], [1, 1323, 900], [1, 1103, 901]], 'outer_group': '64.202', 'outer_hash': 202, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [5160, 126, 127, 367920, 806542], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 29, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 10], [8, 2], [10, 8], [16, 2], [20, 2], [40, 2]], 'representations': {'PC': {'code': 4744006602381113073940507746820810725605727191660030009, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -5, -2, -2, -2, -11, 36, 4636, 4210, 6877, 80, 10511, 1768, 102, 25212, 4219, 124, 15693, 9820]}, 'GLZN': {'d': 2, 'p': 77, 'gens': [20872622, 29218127, 457073, 30983305, 5807055, 19630962, 15522156]}, 'Perm': {'d': 29, 'gens': [1, 427033839765991, 799819538072784, 11728439597242967934246912000, 1181150107543566, 1559624222783880, 327521760645639334802227200000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}:C_{10}\\times Q_{16}', 'transitive_degree': 352, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '5.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [2]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', '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]], 'center_label': '5.1', 'center_order': 5, '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': ['5.1'], '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], [5, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '5.1', 'hash': 1, 'hyperelementary': 5, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 5]], 'label': '5.1', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5', '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': 5, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 5, 'order_factorization_type': 1, 'order_stats': [[1, 1], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 5, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -5]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26329372]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131]}, 'Perm': {'d': 5, 'gens': [96]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [5], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
