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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.26844', 'ambient_counter': 26844, 'ambient_order': 256, 'ambient_tex': 'D_{16}:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 32, 'counter': 60, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '256.26844.4.bd1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.bd1.a1', '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': 4, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '64.190', 'subgroup_hash': 190, 'subgroup_order': 64, 'subgroup_tex': 'D_{16}:C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.26844', 'aut_centralizer_order': 4, 'aut_label': '4.bd1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '128.1674', 'aut_weyl_index': 16, 'centralizer': '128.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.o1.a1', '8.bj1.a1', '8.bl1.a1', '8.bp1.a1', '8.bv1.a1', '8.bx1.a1', '8.ca1.a1'], 'core': '8.o1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [9204, -1, 9515, -1, 9193, -1, 9502, -1], 'generators': [1, 2, 180], 'label': '256.26844.4.bd1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '4.bd1.a1', 'projective_image': '128.2011', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.bd1.a1', 'subgroup_fusion': None, 'weyl_group': '64.250'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 2, 4, 8], 'aut_gens': [[1, 2, 4], [33, 2, 36], [33, 42, 28], [33, 51, 21], [1, 18, 52], [1, 42, 36]], 'aut_group': '256.54439', 'aut_hash': 54439, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 256, 'aut_permdeg': 16, 'aut_perms': [2878807271470, 5411763724098, 2878807273379, 2877362637000, 4391613695298], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 8, 1, 1], [2, 8, 2, 1], [4, 2, 1, 2], [4, 8, 1, 1], [8, 2, 2, 1], [8, 4, 1, 1], [16, 4, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4^2:C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '32.46', 'autcentquo_hash': 46, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 8, 3], [4, 2, 2], [4, 8, 1], [8, 2, 2], [8, 4, 1], [16, 4, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '32.39', 'commutator_count': 1, 'commutator_label': '8.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 190, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 8, 1, 3], [4, 2, 1, 2], [4, 8, 1, 1], [8, 2, 2, 1], [8, 4, 1, 1], [16, 4, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 336, 'exponent': 16, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, 1, 2]], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '8.5', 'hash': 190, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 2, 8], 'inner_gens': [[1, 2, 36], [1, 2, 60], [33, 10, 4]], 'inner_hash': 39, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'C_2\\times D_8', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 6], [4, 2]], 'label': '64.190', 'linC_count': 2, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D16:C2', 'ngens': 3, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 11, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 16, 'number_divisions': 13, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 37, 'number_subgroup_classes': 45, 'number_subgroups': 113, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 27], [4, 12], [8, 8], [16, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 1, 0], 'outer_gens': [[33, 2, 4], [1, 2, 20], [1, 3, 5]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 2], [8, 1]], 'representations': {'PC': {'code': 75558718372060252, 'gens': [1, 2, 3], 'pres': [6, -2, 2, 2, -2, -2, -2, 650, 548, 50, 681, 69, 730, 88]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [110881, 110623, 111745, 2545103, 2802143, 111169]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [4105, 4129, 22781, 4161, 4225, 14525]}, 'Perm': {'d': 16, 'gens': [11212821043200, 6314147152, 1313941668480, 8402167089238, 4097506706057, 1313941673647]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{16}:C_2', 'transitive_degree': 16, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 8, 'aut_gen_orders': [8, 8, 2, 8, 2, 8], 'aut_gens': [[1, 2, 4, 64], [217, 58, 52, 240], [73, 10, 36, 64], [105, 34, 28, 192], [249, 50, 12, 112], [177, 18, 60, 192], [1, 58, 20, 64]], 'aut_group': None, 'aut_hash': 752670059296835666, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 2048, 'aut_permdeg': 24, 'aut_perms': [604271643510767082193548, 330032121757304641480489, 26495789411355375690233, 270413087534405157996092, 479651935898268629133852, 13021951546770499852450], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 2, 1], [2, 8, 1, 2], [2, 16, 2, 1], [4, 2, 1, 2], [4, 4, 2, 2], [4, 8, 1, 2], [4, 16, 2, 1], [8, 2, 2, 2], [8, 4, 1, 2], [8, 8, 2, 1], [8, 16, 1, 2], [16, 2, 4, 1], [16, 4, 2, 1], [16, 4, 4, 1], [16, 8, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4^2.C_2^3.C_2^4', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '128.2194', 'autcentquo_hash': 2194, 'autcentquo_nilpotent': True, 'autcentquo_order': 128, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_4^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 2], [2, 8, 2], [2, 16, 2], [4, 2, 2], [4, 4, 4], [4, 8, 2], [4, 16, 2], [8, 2, 4], [8, 4, 2], [8, 8, 2], [8, 16, 2], [16, 2, 4], [16, 4, 6], [16, 8, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '128.2011', 'commutator_count': 1, 'commutator_label': '16.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 26844, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 2], [2, 8, 1, 2], [2, 16, 1, 2], [4, 2, 1, 2], [4, 4, 1, 4], [4, 8, 1, 2], [4, 16, 1, 2], [8, 2, 2, 2], [8, 4, 1, 2], [8, 8, 1, 2], [8, 16, 1, 2], [16, 2, 4, 1], [16, 4, 2, 1], [16, 4, 4, 1], [16, 8, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 645120, 'exponent': 16, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, 0, 8]], 'familial': False, 'frattini_label': '16.5', 'frattini_quotient': '16.14', 'hash': 26844, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 2, 8, 4], 'inner_gens': [[1, 2, 4, 240], [1, 2, 28, 80], [1, 42, 4, 64], [145, 50, 4, 64]], 'inner_hash': 2011, 'inner_nilpotent': True, 'inner_order': 128, 'inner_split': False, 'inner_tex': 'D_4\\times D_8', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 16], [4, 11]], 'label': '256.26844', 'linC_count': 8, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 2, 'linQ_dim': 16, 'linQ_dim_count': 2, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D16:D4', 'ngens': 4, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 25, 'number_characteristic_subgroups': 53, 'number_conjugacy_classes': 43, 'number_divisions': 32, 'number_normal_subgroups': 105, 'number_subgroup_autclasses': 215, 'number_subgroup_classes': 326, 'number_subgroups': 1015, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 59], [4, 68], [8, 64], [16, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 104, 0], 'outer_gens': [[1, 2, 4, 96], [105, 34, 4, 64], [1, 2, 20, 64]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [120, 5040, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 32, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 5], [8, 1], [16, 2]], 'representations': {'PC': {'code': 167454138931776382468421263645601884, 'gens': [1, 2, 3, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 346, 66, 907, 91, 972, 116, 13446, 2254, 166, 10247, 5135]}, 'Perm': {'d': 32, 'gens': [8864346369514948454174359610742, 135955991562369039230941840932864000, 67986860127283896396511282924227143, 67977995780968282257608325225692856, 50973977447395947166374977137321656, 16985047389861003970253793579100937, 25199055155403427461045987627105862, 8231691320578226211046411712681647]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{16}:D_4', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
