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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1760.282', 'ambient_counter': 282, 'ambient_order': 1760, 'ambient_tex': 'C_4^2:F_{11}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': True, 'core_order': 88, 'counter': 47, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1760.282.20.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '20.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '20.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_{10}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '88.11', 'subgroup_hash': 11, 'subgroup_order': 88, 'subgroup_tex': 'C_2\\times D_{22}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.282', 'aut_centralizer_order': 16, 'aut_label': '20.a1', 'aut_quo_index': 12, 'aut_stab_index': 1, 'aut_weyl_group': '880.118', 'aut_weyl_index': 16, 'centralizer': '220.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '10.a1.a1', '10.b1.a1', '10.b1.b1'], 'contains': ['40.a1.a1', '40.b1.a1', '40.b1.b1', '40.e1.a1', '40.e1.b1', '220.c1.a1'], 'core': '20.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4454, 3601, 4742, 6200, 4625, 6016, 4868, 5874], 'generators': [5, 880, 40, 220], 'label': '1760.282.20.a1.a1', 'mobius_quo': 0, 'mobius_sub': -2, 'normal_closure': '20.a1.a1', 'normal_contained_in': ['4.a1.a1', '10.a1.a1', '10.b1.a1', '10.b1.b1'], 'normal_contains': ['40.a1.a1', '40.b1.a1', '40.b1.b1'], 'normalizer': '1.a1.a1', 'old_label': '20.a1.a1', 'projective_image': '440.42', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '20.a1.a1', 'subgroup_fusion': None, 'weyl_group': '220.7'}
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gps_groups • Show schema
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{'Agroup': True, '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': 660, 'aut_gen_orders': [20, 30, 2, 22], 'aut_gens': [[1, 2, 4], [45, 15, 76], [44, 11, 69], [1, 46, 4], [1, 67, 4]], 'aut_group': '2640.bv', 'aut_hash': 5844094576270439126, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2640, 'aut_permdeg': 15, 'aut_perms': [667250454382, 125786712251, 7, 838021661423], 'aut_phi_ratio': 66.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 11, 4, 1], [11, 2, 5, 1], [22, 2, 15, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times F_{11}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 11, 4], [11, 2, 5], [22, 2, 15]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '22.1', 'commutator_count': 1, 'commutator_label': '11.1', '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': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['22.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 11, 1, 4], [11, 2, 5, 1], [22, 2, 5, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 84, 'exponent': 22, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '88.11', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [1, 2, 11], 'inner_gens': [[1, 2, 4], [1, 2, 84], [1, 10, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 22, 'inner_split': True, 'inner_tex': 'D_{11}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 20]], 'label': '88.11', 'linC_count': 60, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 12, 'linQ_dim': 11, 'linQ_dim_count': 12, 'linR_count': 60, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D22', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 28, 'number_divisions': 12, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 32, 'number_subgroups': 142, 'old_label': None, 'order': 88, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 47], [11, 10], [22, 30]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [10, 2, 5, 2, 15], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 47, 20], [1, 3, 4], [1, 2, 12], [44, 47, 41], [44, 47, 61]], 'outer_group': '120.37', 'outer_hash': 37, 'outer_nilpotent': False, 'outer_order': 120, 'outer_permdeg': 9, 'outer_perms': [90753, 131760, 64, 10800, 5850], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_5\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [10, 4]], 'representations': {'PC': {'code': 361356200009, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -11, 510, 34, 647]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [922111610, 2296050935, 214518602, 2143735230]}, 'Perm': {'d': 15, 'gens': [6267305647, 7, 16, 99712902360]}}, 'schur_multiplier': [2, 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': 'C_2\\times D_{22}', 'transitive_degree': 44, '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': '80.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [10, 20, 20, 20, 20, 20], 'aut_gens': [[1, 10, 440], [921, 410, 440], [261, 1630, 1540], [201, 470, 1320], [1281, 790, 660], [61, 1350, 1320], [121, 1510, 1540]], 'aut_group': None, 'aut_hash': 7025978137286736026, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14080, 'aut_permdeg': 52, 'aut_perms': [56316018020477509132564631157710152861864374482042574076839693640629, 36753262672541908041198110101682800551140683815248857527975620653408, 16854316585691337928783175494783831703743225628279044854369857839939, 34472818104571872106530855691938440589500580736232776447544736524191, 62127613342674992826954680009926209447434762213105025643291984367729, 4288632689031827384536962347355114008344418205091083863729029069668], 'aut_phi_ratio': 22.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 22, 2, 1], [4, 1, 4, 1], [4, 2, 4, 1], [4, 22, 2, 1], [4, 22, 4, 1], [5, 11, 1, 4], [10, 11, 1, 4], [10, 11, 2, 4], [10, 22, 2, 4], [11, 10, 1, 1], [20, 11, 4, 4], [20, 22, 2, 4], [20, 22, 4, 8], [22, 10, 1, 1], [22, 10, 2, 1], [44, 10, 4, 1], [44, 10, 8, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2^3\\times C_{11}:C_5).C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 22, 2], [4, 1, 4], [4, 2, 4], [4, 22, 6], [5, 11, 4], [10, 11, 12], [10, 22, 8], [11, 10, 1], [20, 11, 16], [20, 22, 40], [22, 10, 3], [44, 10, 12]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '220.7', 'commutator_count': 1, 'commutator_label': '22.2', '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': 282, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 22, 1, 2], [4, 1, 2, 2], [4, 2, 2, 2], [4, 22, 1, 2], [4, 22, 2, 2], [5, 11, 4, 1], [10, 11, 4, 3], [10, 22, 4, 2], [11, 10, 1, 1], [20, 11, 8, 2], [20, 22, 4, 2], [20, 22, 8, 4], [22, 10, 1, 3], [44, 10, 2, 2], [44, 10, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 124992, 'exponent': 220, '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': '4.2', 'frattini_quotient': '440.42', 'hash': 282, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 22, 1], 'inner_gens': [[1, 1010, 440], [1201, 10, 440], [1, 10, 440]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 220, 'inner_split': False, 'inner_tex': 'C_2\\times F_{11}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 20], [10, 16]], 'label': '1760.282', 'linC_count': 320, 'linC_degree': 11, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 32, 'linQ_dim': 16, 'linQ_dim_count': 32, 'linR_count': 800, 'linR_degree': 16, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2:F11', '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': 45, 'number_characteristic_subgroups': 25, 'number_conjugacy_classes': 116, 'number_divisions': 36, 'number_normal_subgroups': 71, 'number_subgroup_autclasses': 92, 'number_subgroup_classes': 152, 'number_subgroups': 954, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 47], [4, 144], [5, 44], [10, 308], [11, 10], [20, 1056], [22, 30], [44, 120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1101, 230, 660], [221, 10, 660], [221, 210, 440], [1, 230, 1320], [1101, 450, 660]], 'outer_group': '64.202', 'outer_hash': 202, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [368063, 5040, 368040, 7, 1169406], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 10], [8, 4], [10, 4], [16, 2], [20, 2], [40, 2]], 'representations': {'PC': {'code': 2613249928912493839894439653005569852610300900, 'gens': [1, 3, 6], 'pres': [7, -2, -5, -2, -2, -11, -2, -2, 14, 21212, 3894, 58, 7283, 4210, 80, 2804, 2811, 124]}, 'GLZN': {'d': 2, 'p': 88, 'gens': [1193611, 15673879, 682177, 46173577, 6133329, 30843451, 685345]}, 'Perm': {'d': 23, 'gens': [51212945749253194800, 47991007, 91578257, 107304560072802892800, 135732247, 135732240, 1277293892726538086400]}}, 'schur_multiplier': [2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2:F_{11}', 'transitive_degree': 176, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1, 2], [10, 13], [10, 15]], 'aut_group': '24.5', 'aut_hash': 5, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [127, 857], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 4, 1], [10, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.5', 'autcent_hash': 5, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times S_3', '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], [5, 1, 4], [10, 1, 12]], 'center_label': '20.5', 'center_order': 20, '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', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 1], [10, 1, 4, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '20.5', 'hash': 5, '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, 20]], 'label': '20.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 6, 'linQ_dim': 5, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C10', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 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': 20, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [5, 4], [10, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[10, 13], [10, 15]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [127, 857], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4]], 'representations': {'PC': {'code': 2179, 'gens': [1, 2], 'pres': [3, -2, -2, -5, 16]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706303489327]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [13320, 4001]}, 'Perm': {'d': 9, 'gens': [40320, 720, 96]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], '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_{10}', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
