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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.26539', 'ambient_counter': 26539, 'ambient_order': 256, 'ambient_tex': 'D_4^2:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 16, 'counter': 166, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '256.26539.8.dd1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.dd1.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': 8, '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': '32.28', 'subgroup_hash': 28, 'subgroup_order': 32, 'subgroup_tex': 'C_4:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.26539', 'aut_centralizer_order': 8, 'aut_label': '8.dd1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '64.267', 'aut_weyl_index': 16, 'centralizer': '64.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.d1.a1', '4.bj1.a1', '4.bk1.a1'], 'contains': ['16.e1.a1', '16.cr1.a1', '16.cu1.a1', '16.cy1.a1', '16.da1.a1', '16.ee1.a1', '16.ek1.a1'], 'core': '16.e1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [4466, -1, 3355, -1, 4461, -1, 3357, -1], 'generators': [9, 18, 64], 'label': '256.26539.8.dd1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '8.dd1.a1', 'projective_image': '128.1755', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.dd1.a1', 'subgroup_fusion': None, 'weyl_group': '32.51'}
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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': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 4], 'aut_gens': [[1, 2, 8], [17, 2, 8], [5, 18, 8], [17, 18, 12], [5, 6, 24], [1, 6, 8], [1, 14, 8]], 'aut_group': '128.2163', 'aut_hash': 2163, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 128, 'aut_permdeg': 12, 'aut_perms': [101693656, 7, 182610840, 14923560, 138989520, 248266207], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 4, 2, 1], [4, 2, 2, 2], [4, 4, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', '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, 3], [2, 2, 2], [2, 4, 2], [4, 2, 4], [4, 4, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 28, '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], [2, 4, 1, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 84, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '8.5', 'hash': 28, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[1, 6, 8], [5, 2, 24], [1, 18, 8]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 6]], 'label': '32.28', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 4, 'linQ_dim': 4, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4:D4', 'ngens': 3, 'nilpotency_class': 2, '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': 9, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 14, 'number_divisions': 13, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 47, 'number_subgroups': 74, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15], [4, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [8, 0, 0, 0], 'outer_gens': [[1, 10, 8], [1, 2, 12], [1, 2, 24], [17, 2, 8]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 1]], 'representations': {'PC': {'code': 604570138, 'gens': [1, 2, 4], 'pres': [5, 2, 2, 2, 2, 2, 61, 26, 248, 58]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931074, 26128940, 26483560]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [34876415, 29555524, 28816400, 28233362, 33533700]}, 'Perm': {'d': 8, 'gens': [1567, 13, 5183, 23, 11520]}}, 'schur_multiplier': [2, 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': 'C_4:D_4', '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': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4, 16, 64], [1, 162, 228, 48, 192], [1, 2, 132, 144, 64], [1, 2, 133, 176, 64], [129, 170, 172, 112, 64], [129, 130, 132, 144, 64], [1, 2, 140, 48, 96], [1, 130, 4, 16, 64]], 'aut_group': '1024.djx', 'aut_hash': 8067704780552372617, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 1024, 'aut_permdeg': 32, 'aut_perms': [49264457878633783793149526241604193, 57870890986034814160692201294775968, 74334251835156098185235185621829454, 175651474622941874125406532395489609, 1535750015628560413822207603254, 41507532029449401073767001163945808, 7738666704631841358666045591066832], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 4, 1, 4], [2, 4, 2, 1], [2, 8, 1, 3], [2, 8, 2, 1], [4, 4, 1, 2], [4, 8, 1, 8], [4, 8, 2, 1], [4, 16, 2, 2], [8, 16, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^6: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': '64.202', 'autcentquo_hash': 202, 'autcentquo_nilpotent': True, 'autcentquo_order': 64, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3:D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 4, 6], [2, 8, 5], [4, 4, 2], [4, 8, 10], [4, 16, 4], [8, 16, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '128.1755', 'commutator_count': 1, 'commutator_label': '16.11', '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': 26539, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 4, 1, 6], [2, 8, 1, 5], [4, 4, 1, 2], [4, 8, 1, 10], [4, 16, 1, 4], [8, 16, 1, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1290240, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[8, 1, 2]], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '16.14', 'hash': 26539, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 4, 4, 2], 'inner_gens': [[1, 2, 4, 144, 64], [1, 2, 12, 16, 224], [1, 10, 4, 80, 224], [129, 2, 196, 16, 64], [1, 162, 164, 16, 64]], 'inner_hash': 1755, 'inner_nilpotent': True, 'inner_order': 128, 'inner_split': True, 'inner_tex': 'C_2^4:D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 12], [4, 4], [8, 2]], 'label': '256.26539', 'linC_count': 2, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'D4^2:C2^2', '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': 28, 'number_characteristic_subgroups': 69, 'number_conjugacy_classes': 34, 'number_divisions': 34, 'number_normal_subgroups': 109, 'number_subgroup_autclasses': 528, 'number_subgroup_classes': 663, 'number_subgroups': 2287, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 71], [4, 152], [8, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 130, 4, 16, 64], [129, 2, 4, 16, 64], [1, 2, 5, 16, 192]], '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': 5, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 4], [8, 2]], 'representations': {'PC': {'code': 42190583597510193161213148085371988, 'gens': [1, 2, 3, 5, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 154, 66, 5764, 820, 268, 116, 1941, 6286, 3158, 1374, 166]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [31029334, 6210625, 32713795, 20225089, 5128257, 18750531, 7292993, 11762753]}, 'Perm': {'d': 16, 'gens': [11212821043200, 6227407417, 1313941685176, 2789792421136, 5606234750016, 2789832700823, 5167, 1313941673647]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4^2:C_2^2', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
