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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '2048.crx', 'ambient_counter': 1818, 'ambient_order': 2048, 'ambient_tex': 'D_4^2.C_2^5', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 256, 'counter': 1512, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2048.crx.8._.BAY', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '8.BAY', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': None, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '256.6664', 'subgroup_hash': None, 'subgroup_order': 256, 'subgroup_tex': '(C_4\\times C_8).D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2048.crx', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [129, 130, 1032], 'label': '2048.crx.8._.BAY', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '8.BAY', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8._.BAY', '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': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 8, 32], [193, 206, 72, 224], [1, 18, 8, 160], [173, 62, 8, 104], [85, 70, 24, 184], [1, 130, 8, 32], [129, 158, 216, 160]], 'aut_group': '1024.djj', 'aut_hash': 354343342895453903, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 1024, 'aut_permdeg': 18, 'aut_perms': [732828465862830, 442764646255, 358568163486126, 3449282203323414, 1, 3948499538091606], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 8, 2, 2], [4, 4, 1, 3], [4, 8, 2, 2], [4, 16, 1, 2], [4, 32, 2, 1], [8, 4, 2, 1], [8, 8, 1, 1], [8, 16, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'D_4^2.C_2^4', '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': 8, 'autcentquo_group': '128.928', 'autcentquo_hash': 928, 'autcentquo_nilpotent': True, 'autcentquo_order': 128, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\wr D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 8, 4], [4, 4, 3], [4, 8, 4], [4, 16, 2], [4, 32, 2], [8, 4, 2], [8, 8, 1], [8, 16, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '128.928', 'commutator_count': 1, 'commutator_label': '32.34', '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': 6664, '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, 1], [2, 8, 1, 4], [4, 4, 1, 3], [4, 8, 1, 2], [4, 8, 2, 1], [4, 16, 1, 2], [4, 32, 1, 2], [8, 4, 2, 1], [8, 8, 1, 1], [8, 16, 1, 2], [8, 16, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 5376, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, 0, 4], [8, -1, 1]], 'familial': False, 'frattini_label': '32.34', 'frattini_quotient': '8.5', 'hash': 6664, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 4, 4, 4], 'inner_gens': [[1, 134, 72, 96], [133, 2, 216, 56], [193, 82, 8, 32], [193, 10, 8, 32]], 'inner_hash': 928, 'inner_nilpotent': True, 'inner_order': 128, 'inner_split': True, 'inner_tex': 'C_2\\wr D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 6], [4, 10], [8, 1]], 'label': '256.6664', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': '(C4*C8).D4', 'ngens': 3, 'nilpotency_class': 5, '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': 17, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 25, 'number_divisions': 22, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 129, 'number_subgroup_classes': 197, 'number_subgroups': 735, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 35], [4, 140], [8, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 16], 'outer_gens': [[129, 2, 8, 32], [1, 2, 8, 160], [45, 62, 8, 232]], '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': 4, 'perfect': False, 'permutation_degree': 32, '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, 6], [8, 1], [16, 1]], 'representations': {'PC': {'code': 46034335309393420816380827772044098849650220778, 'gens': [1, 2, 4, 6], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 2145, 41, 2307, 3467, 211, 91, 5764, 2892, 4613, 1357, 1173, 141, 10758, 2254, 2710, 166]}, 'Perm': {'d': 32, 'gens': [16985047389969122135499973715913737, 144187682882947265441986938703872000, 67977995780968282257602718863923200, 59187996939413163037656623791490496, 33988952657421979333960739503507073, 25199044282045187207315527271184142, 25903229683158464676480, 8231691320578226211046411712681647]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_4\\times C_8).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': '64.267', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [4, 4, 8, 8, 4, 4, 4, 4, 4, 4], 'aut_gens': [[1, 2, 8, 32, 128, 512, 1024], [1449, 938, 344, 352, 128, 768, 1680], [669, 122, 1384, 48, 128, 576, 1280], [1721, 506, 920, 32, 128, 768, 1088], [1109, 822, 1016, 96, 592, 144, 1680], [897, 1894, 1304, 368, 128, 512, 1280], [605, 1306, 952, 368, 384, 832, 2000], [669, 314, 696, 112, 592, 400, 1280], [81, 706, 728, 288, 128, 768, 1024], [861, 810, 360, 368, 128, 832, 1280], [397, 1054, 952, 96, 848, 400, 2000]], 'aut_group': None, 'aut_hash': 6905708186991642715, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 4194304, 'aut_permdeg': 192, 'aut_perms': 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2, 3], [4, 4, 4, 1], [4, 8, 2, 2], [4, 8, 4, 2], [4, 8, 8, 1], [4, 8, 16, 1], [4, 16, 4, 3], [4, 32, 16, 1], [8, 8, 8, 2], [8, 16, 16, 1], [8, 32, 8, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2^6\\times C_4^2).C_2^6.C_2^6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 45958743261578691, 'autcent_nilpotent': True, 'autcent_order': 2048, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^9.C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': None, 'autcentquo_hash': 2125128279056734975, 'autcentquo_nilpotent': True, 'autcentquo_order': 2048, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': '(C_2^2\\times D_4).C_2^6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 6], [2, 4, 4], [2, 8, 32], [2, 16, 4], [4, 1, 4], [4, 2, 6], [4, 4, 12], [4, 8, 36], [4, 16, 12], [4, 32, 16], [8, 8, 16], [8, 16, 16], [8, 32, 8]], 'center_label': '8.2', 'center_order': 8, 'central_product': None, 'central_quotient': '256.26531', 'commutator_count': 1, 'commutator_label': '32.34', 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 1818, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': None, 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 6], [2, 4, 1, 4], [2, 8, 1, 32], [2, 16, 1, 4], [4, 1, 2, 2], [4, 2, 1, 6], [4, 4, 1, 8], [4, 4, 2, 2], [4, 8, 1, 28], [4, 8, 2, 4], [4, 16, 1, 12], [4, 32, 1, 16], [8, 8, 1, 16], [8, 16, 1, 16], [8, 32, 1, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': None, 'exponent': 8, 'exponents_of_order': [11], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '32.34', 'frattini_quotient': '64.267', 'hash': 3770300787620698695, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 4, 4, 4, 1, 2, 2], 'inner_gens': [[1, 6, 8, 368, 128, 768, 1024], [5, 2, 696, 368, 128, 832, 1088], [1, 754, 8, 32, 128, 512, 1280], [81, 82, 8, 32, 128, 512, 1024], [1, 2, 8, 32, 128, 512, 1024], [257, 322, 8, 32, 128, 512, 1024], [1, 66, 264, 32, 128, 512, 1024]], 'inner_hash': 26531, 'inner_nilpotent': True, 'inner_order': 256, 'inner_split': False, 'inner_tex': 'D_4^2:C_2^2', 'inner_used': [1, 2, 3, 7], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 64], [2, 48], [4, 48], [8, 16]], 'label': '2048.crx', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'D4^2.C2^5', 'ngens': 6, 'nilpotency_class': 5, 'nilpotent': True, 'normal_counts': [1, 3, 15, 43, 67, 211, 547, 1011, 1443, 651, 63, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 37, 'number_characteristic_subgroups': 70, 'number_conjugacy_classes': 176, 'number_divisions': 168, 'number_normal_subgroups': 4056, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 2048, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 351], [4, 1056], [8, 640]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4, 4, 4, 4, 4, 4, 4, 4], 'outer_gen_pows': [0, 0, 64, 0, 0, 0, 0, 0], 'outer_gens': [[213, 582, 744, 288, 384, 832, 1344], [333, 986, 104, 368, 848, 400, 1280], [157, 190, 360, 352, 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'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^7.C_2^6.C_2', 'pc_rank': None, 'perfect': False, 'permutation_degree': None, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 6, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 64], [2, 48], [4, 48], [16, 8]], 'representations': {'PC': {'code': '509507777355803551960637071618518789296429132412213422312477260773595566344611221177943227', 'gens': [1, 2, 4, 6, 8, 10, 11], 'pres': [11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2816, 133, 56, 15326, 289, 124, 2215, 3766, 928, 24293, 12160, 1611, 192, 24646, 12337, 260, 84489, 45780, 21151, 65845, 38752, 19403]}, 'Perm': {'d': 34, 'gens': [18182333253529098409221653272416255054, 1, 9761332737502392852791118169995237529, 36363251509237843082373384876702363367, 27423221794098218988118128911582930527, 45277098266390244423081420523305433104]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2, 2], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 4, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4^2.C_2^5', 'transitive_degree': None, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 84, 'aut_gen_orders': [3, 3], 'aut_gens': [[1, 2, 4], [4, 5, 3], [2, 4, 1]], 'aut_group': '168.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 7, 'aut_perms': [4361, 244], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 168, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\PSL(2,7)', '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, 7]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, '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', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 8]], 'label': '8.5', 'linC_count': 28, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 28, 'linQ_dim': 3, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 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': 8, 'number_divisions': 8, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 5, 3], [2, 4, 1]], 'outer_group': '168.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 168, 'outer_permdeg': 7, 'outer_perms': [4361, 244], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -2, 2, 2]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16482, 16322, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 13286, 13933]}, 'Perm': {'d': 6, 'gens': [120, 6, 1]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
