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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '320.572', 'ambient_counter': 572, 'ambient_order': 320, 'ambient_tex': 'C_2^4.D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 80, 'counter': 16, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '320.572.4.h1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.h1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '80.19', 'subgroup_hash': 19, 'subgroup_order': 80, 'subgroup_tex': 'C_{10}.D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '320.572', 'aut_centralizer_order': 8, 'aut_label': '4.h1', 'aut_quo_index': 1, 'aut_stab_index': 4, 'aut_weyl_group': '320.1638', 'aut_weyl_index': 32, 'centralizer': '40.a1.a1', 'complements': ['80.q1.a1', '80.t1.a1', '80.q1.b1', '80.k1.a1', '80.r1.a1', '80.k1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['8.c1.a1', '8.d1.a1', '8.e1.b1', '20.j1.a1'], 'core': '4.h1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8321, 8460, 6620, 8387, 8510, 5980, 8077, 8999], 'generators': [4, 8, 16, 160, 64], 'label': '320.572.4.h1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.h1.a1', 'normal_contained_in': ['2.b1.a1'], 'normal_contains': ['8.c1.a1', '8.d1.a1', '8.e1.b1'], 'normalizer': '1.a1.a1', 'old_label': '4.h1.a1', 'projective_image': '80.36', 'quotient_action_image': '2.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '4.h1.a1', 'subgroup_fusion': None, 'weyl_group': '40.13'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [2, 2, 4, 4, 10], 'aut_gens': [[1, 4, 8], [3, 4, 8], [5, 4, 10], [1, 4, 24], [45, 4, 14], [53, 4, 8]], 'aut_group': '640.19303', 'aut_hash': 19303, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 640, 'aut_permdeg': 13, 'aut_perms': [482630400, 1520467200, 482671450, 3612107520, 482671508], 'aut_phi_ratio': 20.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [4, 10, 4, 1], [5, 2, 2, 1], [10, 2, 2, 1], [10, 2, 4, 1], [10, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\wr C_2\\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': 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, 2], [4, 10, 4], [5, 2, 2], [10, 2, 14]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '20.4', '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', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 19, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [4, 10, 2, 2], [5, 2, 2, 1], [10, 2, 2, 3], [10, 2, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 20, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '20.4', 'hash': 19, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 1, 10], 'inner_gens': [[1, 4, 76], [1, 4, 8], [21, 4, 8]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'D_{10}', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 18]], 'label': '80.19', 'linC_count': 32, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 10, 'linR_count': 40, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C10.D4', '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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 26, 'number_divisions': 14, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 22, 'number_subgroup_classes': 34, 'number_subgroups': 74, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [4, 40], [5, 4], [10, 28]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[41, 4, 8], [3, 4, 56], [47, 4, 14]], 'outer_group': '32.25', 'outer_hash': 25, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [5047, 17, 16567], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 4], [8, 2]], 'representations': {'PC': {'code': 27274720257027, 'gens': [1, 3, 4], 'pres': [5, -2, -2, -2, 2, -5, 10, 1523, 58, 1604]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 35907046251, 14536340629, 58658042680, 135202078093]}, 'Perm': {'d': 13, 'gens': [530934487, 1079215920, 1119243600, 1654863840, 37]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}.D_4', 'transitive_degree': 40, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 5], 'aut_gens': [[1, 4, 16, 32], [3, 4, 16, 32], [9, 4, 16, 32], [17, 4, 16, 32], [1, 6, 16, 32], [1, 12, 16, 32], [1, 20, 16, 32], [1, 4, 16, 34], [1, 4, 16, 40], [1, 4, 16, 48], [1, 4, 16, 96], [1, 68, 16, 32]], 'aut_group': None, 'aut_hash': 2398680610296428710, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10240, 'aut_permdeg': 56, 'aut_perms': [12008306972226306869580383042805585899729555123654265691306529843761341061, 11084925978394328198162891175101409698183596612328153858832929844720068533, 696886633476955110859147902439328546666501327009231520399360001480551965, 25731664805377531518780717065941688425587567698898035642339271988836480, 64205894300438162971407199592420521939399635601970860648240681068132480, 12906859754914423838996089513884157761911653871459045936990649739870080, 672229589216569368698350907474364743331423344524466112496399427722019848896, 620520272175403602547558725573847073676488200649320490081711171722019825280, 38790290919560671970440176798255412421290179535640467737673728020197912336, 4357801859380997237286666225995067432014350185681661621857175680, 17181805133908984243325974517219878912775572932988078986492833237911680], 'aut_phi_ratio': 80.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 4, 2, 1], [4, 2, 4, 1], [4, 4, 2, 1], [4, 10, 4, 2], [4, 20, 2, 2], [5, 2, 2, 1], [10, 2, 2, 7], [10, 4, 8, 1], [20, 4, 8, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '512.10494213', 'autcent_hash': 1718285292446712970, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^9', '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, 7], [2, 4, 2], [4, 2, 4], [4, 4, 2], [4, 10, 8], [4, 20, 4], [5, 2, 2], [10, 2, 14], [10, 4, 8], [20, 4, 16]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '40.13', 'commutator_count': 1, 'commutator_label': '20.5', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 572, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 4, 1, 2], [4, 2, 2, 2], [4, 4, 2, 1], [4, 10, 2, 4], [4, 20, 1, 2], [4, 20, 2, 1], [5, 2, 2, 1], [10, 2, 2, 7], [10, 4, 4, 2], [20, 4, 4, 2], [20, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1008, 'exponent': 20, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '40.13', 'hash': 572, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 2, 1, 10], 'inner_gens': [[1, 4, 16, 56], [1, 4, 16, 304], [1, 4, 16, 32], [25, 84, 16, 32]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': True, 'inner_tex': 'C_2\\times D_{10}', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 44], [4, 8]], 'label': '320.572', 'linC_count': 1024, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 64, 'linQ_dim': 10, 'linQ_dim_count': 64, 'linR_count': 240, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4.D10', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 51, 'number_conjugacy_classes': 68, 'number_divisions': 33, 'number_normal_subgroups': 69, 'number_subgroup_autclasses': 150, 'number_subgroup_classes': 190, 'number_subgroups': 590, 'old_label': None, 'order': 320, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 15], [4, 176], [5, 4], [10, 60], [20, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[3, 4, 16, 32], [9, 4, 16, 32], [17, 4, 16, 32], [1, 6, 16, 32], [1, 12, 16, 32], [1, 4, 16, 34], [1, 4, 16, 96]], 'outer_group': '256.56082', 'outer_hash': 56082, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 16, 'outer_perms': [120, 1307674368000, 6227020800, 39916800, 362880, 5040, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 8], [8, 8], [16, 1]], 'representations': {'PC': {'code': 22461072876992194782248963, 'gens': [1, 3, 5, 6], 'pres': [7, -2, -2, -2, -2, 2, 2, -5, 14, 58, 2357, 3211, 124, 3156]}, 'GLZN': {'d': 2, 'p': 40, 'gens': [1104427, 1744417, 1033484, 96801, 1888829, 64321, 1776427]}, 'Perm': {'d': 17, 'gens': [1313945302450, 87091333, 40291223, 23, 5160, 87091223, 23544365644800]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4.D_{10}', 'transitive_degree': 160, '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': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, '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', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
