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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1600.4623', 'ambient_counter': 4623, 'ambient_order': 1600, 'ambient_tex': 'C_{20}^2.C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 800, 'characteristic': False, 'core_order': 1, 'counter': 359, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1600.4623.800.d1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '800.d1.b1', 'outer_equivalence': False, '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': 800, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1600.4623', 'aut_centralizer_order': 2560, 'aut_label': '800.d1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '1.1', 'aut_weyl_index': 20480, 'centralizer': '2.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['160.j1.b1', '160.k1.b1', '160.l1.c1', '160.l1.d1', '400.d1.b1', '400.f1.a1', '400.h1.a1'], 'contains': ['1600.a1.a1'], 'core': '1600.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [1454, -1, 3637, -1, 2180, -1, 6555, -1], 'generators': [44], 'label': '1600.4623.800.d1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '400.d1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '800.d1.b1', 'projective_image': '1600.4623', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '800.d1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', '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, 1]], 'center_label': '2.1', 'center_order': 2, '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': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '160.190', '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, 2, 20, 4, 4, 20, 2, 4], 'aut_gens': [[1, 8, 80], [7, 428, 244], [1123, 472, 1044], [165, 76, 720], [1447, 476, 80], [805, 24, 84], [961, 876, 240], [1125, 1212, 80], [483, 808, 1520], [1281, 24, 560]], 'aut_group': None, 'aut_hash': 5365616513316371285, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20480, 'aut_permdeg': 52, 'aut_perms': [19331532399769353277950245786607054833625574593092245278889989709847, 9803402599661690113915385958398409118287552840913947911306911440871, 63242254782775200459369559671873132453503837516292426171448978554441, 72305301562787451851048528482633064306018666597947779176251176721594, 34812573288599976762830253564194710504689331699836203740783060744305, 47244250151364295718286172499383948229949841045754098509723362062946, 21321773442285812165175009132873268359484137184514850043740809921857, 7088234104301267722429978732534263018954041839454206581702757622406, 64314631475439049304181930873354510561799416915193423686612658713692], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 4, 1], [4, 1, 2, 2], [4, 2, 2, 2], [4, 2, 4, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [8, 5, 8, 1], [8, 10, 4, 1], [8, 10, 8, 1], [10, 1, 4, 3], [10, 2, 2, 3], [10, 2, 8, 3], [10, 2, 16, 1], [10, 4, 8, 1], [10, 4, 32, 1], [20, 1, 8, 2], [20, 2, 4, 2], [20, 2, 8, 2], [20, 2, 16, 3], [20, 4, 4, 2], [20, 4, 8, 1], [20, 4, 16, 2], [20, 4, 32, 1], [40, 5, 32, 1], [40, 10, 16, 1], [40, 10, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_5:(C_2^6.C_2^6)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '512.10494201', 'autcent_hash': 8909733522074432791, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7\\times C_4', '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], [4, 1, 4], [4, 2, 8], [5, 1, 4], [5, 2, 10], [8, 5, 8], [8, 10, 12], [10, 1, 12], [10, 2, 46], [10, 4, 40], [20, 1, 16], [20, 2, 72], [20, 4, 80], [40, 5, 32], [40, 10, 48]], 'center_label': '40.9', 'center_order': 40, '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', '2.1', '5.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4623, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['40.1', 1], ['5.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 3], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [8, 5, 4, 2], [8, 10, 4, 3], [10, 1, 4, 3], [10, 2, 2, 3], [10, 2, 4, 10], [10, 4, 2, 4], [10, 4, 4, 8], [20, 1, 8, 2], [20, 2, 4, 4], [20, 2, 8, 7], [20, 4, 2, 2], [20, 4, 4, 7], [20, 4, 8, 6], [40, 5, 16, 2], [40, 10, 16, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 62496, 'exponent': 40, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '200.50', 'hash': 4623, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 2, 10], 'inner_gens': [[1, 8, 720], [1, 8, 880], [961, 808, 80]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': False, 'inner_tex': 'C_2\\times D_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 160], [2, 200], [4, 40]], 'label': '1600.4623', 'linC_count': 3072, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 704, 'linQ_dim': 14, 'linQ_dim_count': 384, 'linR_count': 640, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C20^2.C2^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 46, 'number_characteristic_subgroups': 66, 'number_conjugacy_classes': 400, 'number_divisions': 85, 'number_normal_subgroups': 154, 'number_subgroup_autclasses': 188, 'number_subgroup_classes': 362, 'number_subgroups': 820, 'old_label': None, 'order': 1600, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 11], [4, 20], [5, 24], [8, 160], [10, 264], [20, 480], [40, 640]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4, 2, 4, 4, 4, 4], 'outer_gen_pows': [961, 0, 0, 1, 320, 0, 0, 0], 'outer_gens': [[1281, 8, 1044], [163, 1228, 720], [1123, 472, 1044], [801, 872, 560], [961, 24, 84], [961, 428, 1360], [165, 828, 244], [485, 460, 1364]], 'outer_group': '512.10493066', 'outer_hash': 4599981025227913554, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 256, 'outer_perms': 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644544102999097950020122593839528408905442481078379524632866354096440803759646586678067662621326989960344803085705235261536658059699449206293893544157716325043938644928444323927815678469621234683758159861270977283268083131647975582285934211640777892656377976709715029258023350026370789981159000135631002204616571753778644123162974234972448609607367834464147723944539214478386828246553418785965089962790670386074041890895727558895693948301624242894375948068481044718363954048816432622663935597522445860710263], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2.C_2^5', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 8, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 21], [8, 29], [16, 18], [32, 3]], 'representations': {'PC': {'code': 677179949020100452497140555849520284106947, 'gens': [1, 4, 6], 'pres': [8, -2, -2, -2, -2, -5, -2, -2, -5, 16, 41, 91, 34565, 5309, 141, 80646, 166, 81927]}, 'GLZN': {'d': 2, 'p': 75, 'gens': [13500032, 21111125, 10267524, 13419061, 20671924, 23100, 18140668, 26075281]}, 'Perm': {'d': 22, 'gens': [51212587272158558463, 121645100408844316, 107047688359772172316, 378005070643200, 51212587272118272000, 18550, 12316, 559198080]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 40], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}^2.C_2^2', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
