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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '800.1159', 'ambient_counter': 1159, 'ambient_order': 800, 'ambient_tex': 'C_{10}^2:C_2^3', 'central': False, 'central_factor': True, 'centralizer_order': 200, 'characteristic': False, 'core_order': 40, 'counter': 58, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '800.1159.20.j1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '20.j1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '20.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{10}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '40.10', 'subgroup_hash': 10, 'subgroup_order': 40, 'subgroup_tex': 'C_5\\times D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '800.1159', 'aut_centralizer_order': 160, 'aut_label': '20.j1', 'aut_quo_index': 1, 'aut_stab_index': 4, 'aut_weyl_group': '32.25', 'aut_weyl_index': 640, 'centralizer': '4.b1', 'complements': ['40.a1', '40.p1', '40.x1'], 'conjugacy_class_count': 4, 'contained_in': ['4.h1', '10.c1', '10.m1'], 'contains': ['40.d1', '40.h1', '100.c1'], 'core': '20.j1', 'coset_action_label': None, 'count': 4, 'diagramx': [9331, 9223, 773, 9815], 'generators': [21, 160, 400, 200], 'label': '800.1159.20.j1', 'mobius_quo': 0, 'mobius_sub': -10, 'normal_closure': '20.j1', 'normal_contained_in': ['4.h1', '10.c1'], 'normal_contains': ['40.d1', '40.h1', '100.c1'], 'normalizer': '1.a1', 'old_label': '20.j1', 'projective_image': '80.51', 'quotient_action_image': '1.1', 'quotient_action_kernel': '20.4', 'quotient_action_kernel_order': 20, 'quotient_fusion': None, 'short_label': '20.j1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4, 4], 'aut_gens': [[1, 2], [1, 22], [1, 34], [11, 2]], 'aut_group': '32.25', 'aut_hash': 25, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [10800, 11529, 5880], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 2, 8, 1], [20, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1], [5, 1, 4], [10, 1, 4], [10, 2, 8], [20, 2, 4]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 2, 4, 2], [20, 2, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 18, 'exponent': 20, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '20.5', 'hash': 10, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 22], [21, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 20], [2, 5]], 'label': '40.10', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 10, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*D4', 'ngens': 4, 'nilpotency_class': 2, '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': 25, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 20, 'old_label': None, 'order': 40, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 5], [4, 2], [5, 4], [10, 20], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [2, 0], 'outer_gens': [[11, 2], [1, 34]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', '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], [2, 1], [4, 4], [8, 1]], 'representations': {'PC': {'code': 1473908227, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -5, 177, 21, 34]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [25038659807093174, 58438750843733785]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [13320, 132, 4001]}, 'Perm': {'d': 9, 'gens': [40320, 90720, 96, 41040]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_4', 'transitive_degree': 20, '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': '80.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 4, 4, 4, 4, 20, 4], 'aut_gens': [[1, 2, 4, 40], [601, 418, 12, 540], [221, 22, 436, 700], [601, 22, 412, 520], [601, 30, 12, 120], [201, 414, 28, 700], [1, 26, 404, 700], [601, 10, 28, 540]], 'aut_group': None, 'aut_hash': 6512940726104921257, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20480, 'aut_permdeg': 32, 'aut_perms': [149127265663149634110354395934917424, 233729249230516510023361550310438339, 132741313979779828925960168081019403, 116235344333148859947337512088973660, 84718130001855968697456988154451903, 214306391578393429714096465036967941, 256026618552006360700225741048103324], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [2, 5, 4, 1], [2, 10, 4, 1], [4, 2, 2, 1], [4, 10, 2, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [10, 1, 4, 1], [10, 1, 8, 1], [10, 2, 2, 1], [10, 2, 4, 1], [10, 2, 8, 1], [10, 2, 16, 2], [10, 4, 8, 1], [10, 4, 32, 1], [10, 5, 16, 1], [10, 10, 16, 1], [20, 2, 8, 1], [20, 4, 4, 1], [20, 4, 16, 1], [20, 10, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_5:(C_2^7.C_2^5)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '512.514818', 'autcent_hash': 3164519744680633539, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4.C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [2, 5, 4], [2, 10, 4], [4, 2, 2], [4, 10, 2], [5, 1, 4], [5, 2, 10], [10, 1, 12], [10, 2, 46], [10, 4, 40], [10, 5, 16], [10, 10, 16], [20, 2, 8], [20, 4, 20], [20, 10, 8]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '40.13', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1159, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['2.1', 1], ['5.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [2, 5, 1, 4], [2, 10, 1, 4], [4, 2, 1, 2], [4, 10, 1, 2], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [10, 1, 4, 3], [10, 2, 2, 3], [10, 2, 4, 10], [10, 4, 2, 4], [10, 4, 4, 8], [10, 5, 4, 4], [10, 10, 4, 4], [20, 2, 4, 2], [20, 4, 2, 2], [20, 4, 4, 4], [20, 10, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6093360, 'exponent': 20, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '400.219', 'hash': 1159, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 2, 5, 2], 'inner_gens': [[1, 2, 4, 440], [1, 2, 36, 40], [1, 10, 4, 40], [401, 2, 4, 40]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': False, 'inner_tex': 'C_2\\times D_{10}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 100], [4, 20]], 'label': '800.1159', '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': 'C10^2:C2^3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 26, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 200, 'number_divisions': 70, 'number_normal_subgroups': 194, 'number_subgroup_autclasses': 148, 'number_subgroup_classes': 526, 'number_subgroups': 1756, 'old_label': None, 'order': 800, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 71], [4, 24], [5, 24], [10, 504], [20, 176]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 2, 4, 2, 4, 2], 'outer_gen_pows': [0, 0, 0, 42, 0, 2], 'outer_gens': [[1, 422, 4, 700], [1, 402, 404, 360], [1, 422, 404, 380], [201, 402, 12, 40], [221, 2, 4, 380], [1, 2, 12, 40]], 'outer_group': '512.7532583', 'outer_hash': 238240055111544898, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 14, 'outer_perms': [40280088, 19678982400, 19639070766, 6267744001, 6268106880, 368047], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^3:C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4], [4, 24], [8, 22], [16, 4]], 'representations': {'PC': {'code': 43306621899510252391619166403, 'gens': [1, 2, 3, 5], 'pres': [7, -2, -2, -2, -5, -2, -2, -5, 387, 58, 458, 15404, 102, 124]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [898450, 35962, 826574, 359380, 794398, 35960, 28083]}, 'Perm': {'d': 16, 'gens': [127, 1, 180583603201, 1313901388800, 43948800, 2789705318400, 966]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_{10}^2:C_2^3', 'transitive_degree': 80, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 10], 'aut_gens': [[1, 2], [1, 6], [7, 2]], 'aut_group': '40.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 40, 'aut_permdeg': 7, 'aut_perms': [169, 2929], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_5', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 2], [5, 2, 2], [10, 2, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', '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': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['10.1', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [5, 2, 2, 1], [10, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 1, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '20.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 18], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 4]], 'label': '20.4', 'linC_count': 2, 'linC_degree': 2, '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': 'D10', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 8, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 22, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 11], [5, 4], [10, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [1, 0], 'outer_gens': [[1, 6], [11, 2]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 2]], 'representations': {'PC': {'code': 2624387, 'gens': [1, 2], 'pres': [3, -2, -2, -5, 109, 16, 146]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7430093, 16717348]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504, 501]}, 'Perm': {'d': 7, 'gens': [126, 1, 966]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{10}', 'transitive_degree': 10, 'wreath_data': None, 'wreath_product': False}
