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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1344.9365', 'ambient_counter': 9365, 'ambient_order': 1344, 'ambient_tex': 'C_{21}:D_4^2', 'central': False, 'central_factor': False, 'centralizer_order': 336, 'characteristic': False, 'core_order': 2, 'counter': 816, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.9365.336.y1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '336.y1.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': 336, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.9365', 'aut_centralizer_order': 16128, 'aut_label': '336.y1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '2.1', 'aut_weyl_index': 32256, 'centralizer': '4.i1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['48.y1.a1', '112.y1.a1', '168.b1.a1', '168.br1.a1', '168.bs1.a1'], 'contains': ['672.c1.a1', '672.e1.a1'], 'core': '672.c1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': None, 'generators': [24600799, 42082055], 'label': '1344.9365.336.y1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '168.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.j1.a1', 'old_label': '336.y1.a1', 'projective_image': '672.1155', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '336.y1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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, 3]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': 42, 'aut_gen_orders': [6, 6, 6, 6, 6, 6, 6, 2, 6, 6, 2], 'aut_gens': [[8005075, 9290415, 34331655, 42082055, 49644328], [49492325, 33984105, 34136775, 42082055, 45495365], [8005075, 9290415, 21689760, 42082055, 45195527], [49495349, 9090495, 9140769, 42082055, 45495365], [24599807, 9090495, 21584760, 42082055, 45195527], [24601823, 9090495, 34284195, 42082055, 45195527], [49496357, 9090495, 9391173, 42082055, 12004075], [8005075, 9290415, 34284195, 42082055, 12303913], [8004067, 9090495, 8996373, 42082055, 45495365], [32901625, 34184025, 34283187, 42082055, 45495365], [8005075, 34184025, 34333671, 42082055, 12004075], [32899609, 9090495, 8996373, 42082055, 12303913]], 'aut_group': None, 'aut_hash': 1198606165070980440, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 64512, 'aut_permdeg': 48, 'aut_perms': [5575169682352479805289414284406246191237686824965591273855898, 10613522557453430738608444656960984014744870244230160608687268, 1536170043698950066927975052589931377484078566147179361649257, 2849272994315390839974790759863614818534775008779041943751348, 9408355342462261033234001511389715330815300661184954278959596, 5455506704454124660588420596240858117261459837373074114119134, 9057541281846339398783382700892792538496661223313587260439921, 2042343138547511750123003427549039585393309587508326019519967, 7192918698412496113223393299682204028991507875152174649981380, 8025393496851542461318863708427011682038113455447942753289689, 1188837559990447840884762079724699758852022495276264245454142], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 4, 1, 1], [2, 12, 1, 1], [2, 14, 2, 2], [2, 28, 1, 1], [2, 42, 2, 1], [2, 84, 1, 1], [3, 2, 1, 1], [4, 4, 1, 1], [4, 6, 2, 1], [4, 12, 1, 2], [4, 14, 2, 1], [4, 84, 1, 2], [6, 2, 1, 3], [6, 4, 2, 2], [6, 14, 4, 1], [6, 28, 2, 2], [7, 2, 3, 1], [12, 4, 2, 1], [12, 28, 2, 1], [14, 2, 3, 3], [14, 4, 6, 1], [14, 8, 3, 1], [14, 24, 3, 1], [21, 4, 3, 1], [28, 8, 3, 1], [28, 12, 6, 2], [28, 24, 3, 1], [42, 4, 3, 3], [42, 8, 6, 2], [84, 8, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^8\\times S_3\\times F_7', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 1], [2, 12, 1], [2, 14, 4], [2, 28, 1], [2, 42, 2], [2, 84, 1], [3, 2, 1], [4, 4, 1], [4, 6, 2], [4, 12, 2], [4, 14, 2], [4, 84, 2], [6, 2, 3], [6, 4, 4], [6, 14, 4], [6, 28, 4], [7, 2, 3], [12, 4, 2], [12, 28, 2], [14, 2, 9], [14, 4, 6], [14, 8, 3], [14, 24, 3], [21, 4, 3], [28, 8, 3], [28, 12, 12], [28, 24, 3], [42, 4, 9], [42, 8, 12], [84, 8, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '336.219', 'commutator_count': 1, 'commutator_label': '84.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 9365, '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, 1], [2, 12, 1, 1], [2, 14, 1, 4], [2, 28, 1, 1], [2, 42, 1, 2], [2, 84, 1, 1], [3, 2, 1, 1], [4, 4, 1, 1], [4, 6, 1, 2], [4, 12, 1, 2], [4, 14, 1, 2], [4, 84, 1, 2], [6, 2, 1, 3], [6, 4, 1, 2], [6, 4, 2, 1], [6, 14, 2, 2], [6, 28, 1, 2], [6, 28, 2, 1], [7, 2, 3, 1], [12, 4, 2, 1], [12, 28, 1, 2], [14, 2, 3, 3], [14, 4, 3, 2], [14, 8, 3, 1], [14, 24, 3, 1], [21, 4, 3, 1], [28, 8, 3, 1], [28, 12, 3, 2], [28, 12, 6, 1], [28, 24, 3, 1], [42, 4, 3, 3], [42, 8, 3, 2], [42, 8, 6, 1], [84, 8, 6, 1]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 14938560, 'exponent': 84, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '336.219', 'hash': 9365, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 2, 14, 1, 6], 'inner_gens': [[8005075, 9290415, 9142785, 42082055, 49644328], [8005075, 9290415, 34331655, 42082055, 33048560], [32897593, 9290415, 34331655, 42082055, 45195527], [8005075, 9290415, 34331655, 42082055, 49644328], [8005075, 9090495, 34278147, 42082055, 49644328]], 'inner_hash': 219, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': True, 'inner_tex': 'D_6\\times D_{14}', 'inner_used': [1, 2, 3, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 48], [4, 47], [8, 6]], 'label': '1344.9365', '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': 'C21:D4^2', 'ngens': 8, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 48, 'number_characteristic_subgroups': 120, 'number_conjugacy_classes': 117, 'number_divisions': 61, 'number_normal_subgroups': 158, 'number_subgroup_autclasses': 600, 'number_subgroup_classes': 856, 'number_subgroups': 8564, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 275], [3, 2], [4, 236], [6, 190], [7, 6], [12, 64], [14, 138], [21, 12], [28, 240], [42, 132], [84, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 2, 6], 'outer_gen_pows': [592705, 592705, 592705, 592705, 592705, 592705], 'outer_gens': [[8005075, 9290415, 34331655, 42082055, 12303913], [32898601, 9290415, 34331655, 42082055, 49644328], [49494341, 9290415, 34331655, 42082055, 49644328], [8005075, 34184025, 34331655, 42082055, 49644328], [8005075, 33984105, 34331655, 42082055, 49644328], [8005075, 9290415, 34329639, 42082055, 49644328]], 'outer_group': '192.1543', 'outer_hash': 1543, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 15, 'outer_perms': [720, 87178291200, 479001600, 3628800, 40320, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\\times C_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 7], [6, 8], [8, 1], [12, 8], [24, 5]], 'representations': {'PC': {'code': 618103991748835845355069116661940799852601, 'gens': [1, 2, 3, 6, 7], 'pres': [8, 2, 2, 2, 2, 7, 2, 2, 3, 2594, 66, 3331, 91, 3844, 10990, 17270, 166, 14359]}, 'GLZN': {'d': 2, 'p': 84, 'gens': [593713, 41391699, 1042546, 16498173, 17188445, 592745, 744451, 21883632]}, 'Perm': {'d': 18, 'gens': [21010447180807, 362893, 1583, 5176, 11520, 23, 3991680, 397707843456000]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{21}:D_4^2', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
