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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.11600', 'ambient_counter': 11600, 'ambient_order': 256, 'ambient_tex': 'C_4^2.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 32, 'counter': 55, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '256.11600.4.z1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.z1.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': 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': '64.133', 'subgroup_hash': 133, 'subgroup_order': 64, 'subgroup_tex': 'D_4.D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.11600', 'aut_centralizer_order': None, 'aut_label': '4.z1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '64.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['8.d1.b1', '8.u1.a1', '8.bb1.a1', '8.bl1.b1', '8.bm1.a1', '8.cg1.b1', '8.ch1.a1'], 'core': '8.d1.b1', 'coset_action_label': None, 'count': 2, 'diagramx': [941, -1, 9244, -1, 940, -1, 9260, -1], 'generators': [33, 74, 16], 'label': '256.11600.4.z1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.a1', 'old_label': '4.z1.b1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.z1.b1', 'subgroup_fusion': None, 'weyl_group': None}
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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, 2, 4], 'aut_gens': [[1, 2, 16, 32], [1, 34, 16, 32], [1, 2, 48, 32], [33, 2, 16, 32], [1, 2, 24, 32], [1, 10, 16, 32], [1, 6, 24, 32], [13, 2, 48, 32]], 'aut_group': '256.56083', 'aut_hash': 56083, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 256, 'aut_permdeg': 20, 'aut_perms': [3768968421792296, 61391729505859592, 1344892980246134581, 40866443034990152, 2450807383475816, 356977849161323, 2379437827792543260], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 4, 2, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 2, 1], [4, 8, 1, 2], [8, 4, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times C_2^5', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '4.2', 'autcentquo_hash': 2, 'autcentquo_nilpotent': True, 'autcentquo_order': 4, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 3], [4, 2, 4], [4, 4, 2], [4, 8, 2], [8, 4, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '16.11', 'commutator_count': 1, 'commutator_label': '8.2', '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': 133, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 3], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 2], [4, 8, 1, 2], [8, 4, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 336, 'exponent': 8, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '8.5', 'hash': 133, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2, 1], 'inner_gens': [[1, 6, 24, 32], [13, 2, 48, 32], [9, 34, 16, 32], [1, 2, 16, 32]], 'inner_hash': 11, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'C_2\\times D_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 10], [4, 1]], 'label': '64.133', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 10, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4.D4', 'ngens': 3, 'nilpotency_class': 3, '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': 14, 'number_characteristic_subgroups': 25, 'number_conjugacy_classes': 19, 'number_divisions': 16, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 61, 'number_subgroup_classes': 76, 'number_subgroups': 137, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15], [4, 32], [8, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 2, 48, 32], [33, 2, 16, 32], [1, 2, 24, 32], [1, 10, 16, 32]], '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': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 1], [8, 1]], 'representations': {'PC': {'code': 1197402799214950, 'gens': [1, 2, 5, 6], 'pres': [6, 2, 2, 2, 2, 2, 2, 73, 31, 218, 50, 724, 730]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [111169, 1734447, 1880081, 111745, 1071039, 1034049]}, 'Perm': {'d': 20, 'gens': [150232650287264041, 275481921337343407, 406634628889772776, 534602333071628760, 663089032007500327, 663089032007500320]}}, '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': 'D_4.D_4', 'transitive_degree': 32, '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': [4, 2, 4, 4, 4, 2, 4, 4, 2], 'aut_gens': [[1, 2, 4, 16, 32], [141, 70, 4, 24, 168], [129, 198, 4, 24, 108], [9, 206, 4, 24, 160], [9, 198, 4, 152, 168], [25, 158, 132, 16, 32], [13, 10, 12, 24, 104], [21, 82, 140, 16, 104], [5, 78, 4, 16, 108], [129, 138, 12, 144, 232]], 'aut_group': None, 'aut_hash': 5328367072911564919, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 48, 'aut_perms': [7796868690696349433027078814596048623644270046814193481499277, 9834522311818573446375827611925220339916196992846881290555346, 6472471230130134064939573407861485540131335006778220322003188, 1410037007400201449924974039122377940949574551820856728109204, 11265996766834788923669593488210767483269013387772355167789321, 6564336344167622835868935381106758516724681409248392326135508, 11342336609336214006762695507119092651865340640152800031411452, 6305730420453678308057471645816309977575107703265115648952954, 10262663739127075328094928776205799552319196456596599893791078], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 16, 1, 1], [4, 2, 2, 3], [4, 4, 1, 3], [4, 4, 4, 1], [4, 8, 2, 5], [4, 16, 1, 1], [8, 8, 2, 2], [8, 8, 4, 1], [8, 16, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '32.51', 'autcentquo_hash': 51, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 1], [2, 16, 1], [4, 2, 6], [4, 4, 7], [4, 8, 10], [4, 16, 1], [8, 8, 8], [8, 16, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '64.202', 'commutator_count': 1, 'commutator_label': '16.2', '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': 11600, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 16, 1, 1], [4, 2, 1, 4], [4, 2, 2, 1], [4, 4, 1, 7], [4, 8, 1, 10], [4, 16, 1, 1], [8, 8, 1, 2], [8, 8, 2, 3], [8, 16, 1, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 161280, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.2', 'frattini_quotient': '16.14', 'hash': 11600, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 2, 4], 'inner_gens': [[1, 2, 12, 16, 44], [1, 2, 4, 24, 100], [9, 2, 4, 16, 32], [1, 10, 4, 16, 160], [5, 206, 4, 144, 32]], 'inner_hash': 202, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': False, 'inner_tex': 'C_2^3:D_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 12]], 'label': '256.11600', 'linC_count': 48, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 16, 'linQ_dim': 16, 'linQ_dim_count': 20, 'linR_count': 20, 'linR_degree': 16, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.C2^4', 'ngens': 4, 'nilpotency_class': 3, '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': 23, 'number_characteristic_subgroups': 49, 'number_conjugacy_classes': 40, 'number_divisions': 36, 'number_normal_subgroups': 117, 'number_subgroup_autclasses': 257, 'number_subgroup_classes': 438, 'number_subgroups': 983, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 23], [4, 136], [8, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [68, 68, 68, 68, 0, 0], 'outer_gens': [[137, 70, 4, 24, 32], [1, 70, 4, 24, 168], [137, 70, 4, 152, 168], [137, 70, 4, 16, 168], [137, 10, 4, 16, 32], [29, 150, 132, 144, 168]], 'outer_group': '128.2163', 'outer_hash': 2163, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [11520, 5183, 83467560, 16687, 87091200, 40291213], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 5, '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, 12], [4, 4], [8, 4]], 'representations': {'PC': {'code': 11345503335735972004438176298615380327, 'gens': [1, 2, 3, 5, 6], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 64, 1097, 290, 66, 492, 2117, 2413, 517, 141, 4038, 5614, 166]}, 'Perm': {'d': 32, 'gens': [1116228929739421407292540204796143, 9056447812586849034103150008427952, 17801279514487712170724636244792984, 1600677241973449643540086399573039, 25783559727972389526085042738740997, 25783559727972389526082500319586516, 34546900871921004857855975536994516, 34546900871921004857851392319488000]}}, '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': 'C_4^2.C_2^4', 'transitive_degree': 128, 'wreath_data': None, 'wreath_product': False}
