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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '512.514818', 'ambient_counter': 514818, 'ambient_order': 512, 'ambient_tex': 'C_2^4.C_2^5', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 32, 'counter': 43, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '512.514818.8.t1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.t1', 'outer_equivalence': True, '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': '64.58', 'subgroup_hash': 58, 'subgroup_order': 64, 'subgroup_tex': 'C_2^2:C_4^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '512.514818', 'aut_centralizer_order': None, 'aut_label': '8.t1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '16.f1', 'complements': None, 'conjugacy_class_count': 84, 'contained_in': ['4.c1', '4.n1'], 'contains': ['16.a1', '16.v1', '16.x1', '16.bb1', '16.bd1'], 'core': '16.a1', 'coset_action_label': None, 'count': 168, 'diagramx': None, 'generators': [9, 4, 72], 'label': '512.514818.8.t1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1', 'old_label': '8.t1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.t1', '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': '32.21', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4, 4, 2, 4], 'aut_gens': [[2741, 1571, 801, 2252], [1715, 2821, 801, 488], [3767, 3623, 801, 3454], [1939, 3847, 801, 1226], [2965, 769, 801, 2252], [2741, 545, 801, 492]], 'aut_group': '2048.cpi', 'aut_hash': 6814437696827469544, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 2048, 'aut_permdeg': 32, 'aut_perms': [34025539397809565181284216428752161, 256716403843510032728629528841962334, 159128753903412561114427833673253951, 136924631690888828569719668468851278, 252726928842212536369875894163276560], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 4, 1], [2, 2, 4, 1], [4, 1, 8, 1], [4, 2, 4, 1], [4, 2, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^5.D_4^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '1024.djl', 'autcent_hash': 7296783744228449167, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^9.C_2', '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, 7], [2, 2, 4], [4, 1, 8], [4, 2, 20]], 'center_label': '16.10', 'center_order': 16, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', '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': 58, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.3', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 4], [4, 1, 2, 4], [4, 2, 2, 10]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 42, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '8.5', 'hash': 58, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 1, 2], 'inner_gens': [[2741, 1571, 801, 2252], [2741, 1571, 801, 200], [2741, 1571, 801, 2252], [2741, 1795, 801, 2252]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8]], 'label': '64.58', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2:C4^2', 'ngens': 3, 'nilpotency_class': 2, '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': 9, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 40, 'number_divisions': 26, 'number_normal_subgroups': 73, 'number_subgroup_autclasses': 43, 'number_subgroup_classes': 129, 'number_subgroups': 185, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15], [4, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4, 2, 4], 'outer_gen_pows': [513, 513, 513, 513, 513], 'outer_gens': [[1715, 2821, 801, 488], [3767, 3623, 801, 3454], [1939, 3623, 801, 1226], [2965, 769, 801, 2252], [2741, 545, 801, 492]], 'outer_group': '512.356671', 'outer_hash': 7280715051136904412, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 32, 'outer_perms': [76644047674872828080861420160, 9285461653148922401139366848679935, 16473215406254102334135904143528837, 26047943757650645658756373124487620, 34546946493677883748704733665473997], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5.C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 4, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 2]], 'representations': {'PC': {'code': 17263788049, 'gens': [1, 3, 4, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 12, 196, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58438762853764568, 91655858505664348, 125101750005088214]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 122297461254, 30594431879, 30704310001, 30590220001, 115200433752]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [2853, 1827, 545, 1539, 88, 1114]}, 'Perm': {'d': 12, 'gens': [47179567, 11653, 80196480, 23, 127370880, 5183]}}, 'schur_multiplier': [2, 2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2:C_4^2', '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': '64.260', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 168, 'aut_gen_orders': [14, 14, 14, 6], 'aut_gens': [[1, 4, 8, 32, 64, 128, 256], [179, 272, 238, 288, 308, 192, 196], [3, 464, 382, 230, 340, 438, 324], [323, 150, 42, 292, 50, 212, 146], [161, 310, 216, 210, 50, 306, 146]], 'aut_group': '2560.a', 'aut_hash': 9183823824299217648, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 176160768, 'aut_permdeg': 128, 'aut_perms': [368230308095268609946141483910075341982894291749683402605373610053638086092290814359478379096676071081616980096481497799624763785784300280178391207042871001135909198881546961855566408123153376375934592649676324184917, 190751465858985467167344287083983790253841470014141074685137843060156417896759747160863861129813132080589969223548871844495345572197082741204181477921920514047593893178392446282869336107516148062651446202284095976337, 17862399125505898746396405240025775409796103231792711009928544044312117545293549560981877116338896478228266166247701981256130907071524912725572749745642857610499125344488792081564029421291485276529593579399506466231, 141970270154644756907984945995177319239323241155414279886391710087303107404254268434762993092964868086269997184625076677239599260034524212589917329880905086520270487655903914338161520515165805789801276689953829749045], 'aut_phi_ratio': 688128.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 7, 2], [2, 2, 56, 1], [2, 8, 2, 1], [4, 1, 16, 1], [4, 2, 56, 1], [4, 8, 2, 1], [4, 8, 14, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_2^{16}.\\PSL(2,7)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': None, 'autcent_hash': 5043351970803840529, 'autcent_nilpotent': True, 'autcent_order': 1048576, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{20}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '168.42', 'autcentquo_hash': 42, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PSL(2,7)', 'cc_stats': [[1, 1, 1], [2, 1, 15], [2, 2, 56], [2, 8, 2], [4, 1, 16], [4, 2, 56], [4, 8, 30]], 'center_label': '32.45', 'center_order': 32, 'central_product': True, 'central_quotient': '16.14', 'commutator_count': 1, 'commutator_label': '8.5', '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', '2.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 514818, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['128.1578', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [2, 2, 1, 56], [2, 8, 1, 2], [4, 1, 2, 8], [4, 2, 2, 28], [4, 8, 1, 14], [4, 8, 2, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 59520, 'exponent': 4, 'exponents_of_order': [9], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.14', 'frattini_quotient': '32.51', 'hash': 514818, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'inner_gens': [[1, 20, 24, 128, 256, 32, 64], [17, 4, 24, 32, 64, 128, 256], [17, 20, 8, 128, 256, 32, 64], [161, 4, 168, 32, 64, 128, 256], [321, 4, 328, 32, 64, 128, 256], [161, 4, 168, 32, 64, 128, 256], [321, 4, 328, 32, 64, 128, 256]], 'inner_hash': 14, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'C_2^4', 'inner_used': [1, 2, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 64], [2, 112]], 'label': '512.514818', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4.C2^5', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, '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': 11, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 176, 'number_divisions': 132, 'number_normal_subgroups': 1706, 'number_subgroup_autclasses': 194, 'number_subgroup_classes': 31452, 'number_subgroups': 62754, 'old_label': None, 'order': 512, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 143], [4, 368]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [6, 4, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[19, 384, 300, 422, 22, 246, 6], [483, 466, 382, 326, 100, 342, 404], [483, 244, 234, 372, 6, 452, 22]], 'outer_group': '2560.a', 'outer_hash': 4855942693118498495, 'outer_nilpotent': False, 'outer_order': 11010048, 'outer_permdeg': 40, 'outer_perms': [298885680521780916109716773774661055136051024776, 268000402017105790194442091336104775446991713816, 5259741648609461899342418870639042968873470863], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^{16}.\\PSL(2,7)', 'pc_rank': None, 'perfect': False, 'permutation_degree': None, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 4], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 72], [4, 28]], 'representations': {'PC': {'code': '28656566897152266552757325309014853917811471881', 'gens': [1, 3, 4, 6, 7, 8, 9], 'pres': [9, 2, 2, 2, 2, 2, 2, 2, 2, 2, 18, 542, 867, 237, 102, 6917, 896, 16134, 2049, 2311, 322, 5192, 683]}}, 'schur_multiplier': [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, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4.C_2^5', 'transitive_degree': 64, 'wreath_data': None, 'wreath_product': False}
