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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '288.874', 'ambient_counter': 874, 'ambient_order': 288, 'ambient_tex': 'C_4.\\SOPlus(4,2)', 'central': True, 'central_factor': False, 'centralizer_order': 288, 'characteristic': True, 'core_order': 2, 'counter': 96, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '288.874.144.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '144.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '144.186', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 186, 'quotient_metabelian': False, 'quotient_nilpotent': False, 'quotient_order': 144, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'S_3^2:C_2^2', 'simple': True, 'solvable': True, 'special_labels': ['Z', 'Phi', 'U1'], 'split': False, 'standard_generators': False, 'stem': True, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '288.874', 'aut_centralizer_order': 1152, 'aut_label': '144.a1', 'aut_quo_index': 4, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 1152, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['48.a1.a1', '48.b1.a1', '72.a1.a1', '72.b1.a1', '72.c1.a1', '72.d1.a1', '72.e1.a1', '72.e1.b1', '72.f1.a1'], 'contains': ['288.a1.a1'], 'core': '144.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5458, 6553, 4850, 2735, 4914, 6737, 5173, 2735], 'generators': [48], 'label': '288.874.144.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '144.a1.a1', 'normal_contained_in': ['16.a1.a1', '72.a1.a1'], 'normal_contains': ['288.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '144.a1.a1', 'projective_image': '144.186', 'quotient_action_image': '1.1', 'quotient_action_kernel': '144.186', 'quotient_action_kernel_order': 144, 'quotient_fusion': None, 'short_label': '144.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', '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, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], '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, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 3, 3], 'aut_gens': [[1, 2, 8, 96], [53, 54, 88, 160], [25, 26, 56, 96], [1, 54, 104, 192], [1, 50, 56, 96], [1, 50, 8, 96], [49, 50, 8, 96], [49, 2, 40, 192], [161, 98, 8, 96], [1, 66, 8, 96]], 'aut_group': '1152.136369', 'aut_hash': 6567885139531058565, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1152, 'aut_permdeg': 36, 'aut_perms': [142695287734238585752948104449008407378859, 302376765474778466381869170350308949563335, 8138021250071352423898191197523198300084, 7982090940244205974602804661817727264870, 35218948777764081742248144574928268002, 59656521782127484266350361695140405681021, 59777759361044415809405009807239931962938, 281085824281153011877567320588828681681090, 1969157356960550789824224570360842606460], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 12, 1, 1], [2, 18, 1, 1], [3, 4, 1, 2], [4, 2, 1, 1], [4, 12, 1, 1], [4, 12, 2, 1], [4, 18, 1, 1], [6, 4, 1, 2], [6, 12, 2, 1], [8, 36, 2, 1], [12, 4, 2, 1], [12, 8, 1, 1], [12, 12, 2, 1], [12, 24, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6^2:C_2^3', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '144.186', 'autcentquo_hash': 186, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^2:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 12, 1], [2, 18, 1], [3, 4, 2], [4, 2, 1], [4, 12, 3], [4, 18, 1], [6, 4, 2], [6, 12, 2], [8, 36, 2], [12, 4, 2], [12, 8, 1], [12, 12, 2], [12, 24, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '144.186', 'commutator_count': 1, 'commutator_label': '36.7', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 874, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 12, 1, 1], [2, 18, 1, 1], [3, 4, 1, 2], [4, 2, 1, 1], [4, 12, 1, 3], [4, 18, 1, 1], [6, 4, 1, 2], [6, 12, 2, 1], [8, 36, 1, 2], [12, 4, 2, 1], [12, 8, 1, 1], [12, 12, 2, 1], [12, 24, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6720, 'exponent': 24, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 4], [8, -1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '144.186', 'hash': 874, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 6, 3], 'inner_gens': [[1, 6, 280, 96], [53, 2, 280, 160], [113, 114, 8, 96], [1, 34, 8, 96]], 'inner_hash': 186, 'inner_nilpotent': False, 'inner_order': 144, 'inner_split': True, 'inner_tex': 'S_3^2:C_2^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 2], [4, 13], [8, 1]], 'label': '288.874', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 9, 'linQ_dim': 12, 'linQ_dim_count': 8, 'linR_count': 2, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C4.SO+(4,2)', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 24, 'number_divisions': 21, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 84, 'number_subgroup_classes': 99, 'number_subgroups': 496, 'old_label': None, 'order': 288, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 31], [3, 8], [4, 56], [6, 32], [8, 72], [12, 88]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [24, 0, 0], 'outer_gens': [[25, 26, 40, 192], [1, 2, 88, 192], [49, 2, 8, 96]], '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': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 9], [8, 1], [16, 1]], 'representations': {'PC': {'code': 11200148761451766677821756303543887847282371607129987, 'gens': [1, 2, 4, 7], 'pres': [7, -2, -2, -2, -2, -2, -3, 3, 336, 85, 36, 1094, 268, 7843, 3930, 297, 80, 6164, 3091, 718, 102, 10757, 5388, 691, 3933, 2372]}, 'Perm': {'d': 22, 'gens': [51347038473976705002, 6405453993010610, 4497833883397, 378101546005063885, 7212929033498, 107425784095838208000, 109852283730309120000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_4.\\SOPlus(4,2)', 'transitive_degree': 48, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [2, 8, 6, 6], 'aut_gens': [[8156, 8101, 13286, 15473, 7433, 13933], [7427, 7372, 13286, 16931, 13994, 14662], [8914, 8830, 13286, 19118, 15502, 17578], [6698, 8101, 13286, 15473, 14044, 16120], [14770, 8101, 13286, 15473, 8891, 14662]], 'aut_group': '576.8299', 'aut_hash': 8299, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 576, 'aut_permdeg': 13, 'aut_perms': [134996521, 199721537, 3557479081, 607113367], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 4, 1], [2, 9, 1, 2], [3, 4, 2, 1], [4, 18, 2, 1], [6, 4, 2, 1], [6, 12, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2:\\SD_{16}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '144.182', 'autcentquo_hash': 182, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 4], [2, 9, 2], [3, 4, 2], [4, 18, 2], [6, 4, 2], [6, 12, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '72.40', 'commutator_count': 1, 'commutator_label': '18.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 186, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['72.40', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 4], [2, 9, 1, 2], [3, 4, 1, 2], [4, 18, 1, 2], [6, 4, 1, 2], [6, 12, 1, 4]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 1680, 'exponent': 12, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '144.186', 'hash': 186, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3, 1, 3, 4, 2], 'inner_gens': [[8156, 7372, 13286, 16931, 10399, 13933], [6698, 8101, 13286, 15473, 12536, 13204], [8156, 8101, 13286, 15473, 7433, 13933], [7427, 8101, 13286, 15473, 6704, 18307], [9667, 9559, 13286, 16931, 7433, 17578], [8156, 7372, 13286, 17660, 11078, 13933]], 'inner_hash': 40, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': False, 'inner_tex': '\\SOPlus(4,2)', 'inner_used': [1, 2, 4, 5], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 2], [4, 8]], 'label': '144.186', 'linC_count': 4, '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': False, 'metacyclic': False, 'monomial': True, 'name': 'S3^2:C2^2', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 18, 'number_divisions': 18, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 86, 'number_subgroups': 414, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 43], [3, 8], [4, 36], [6, 56]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [6643, 6643], 'outer_gens': [[16175, 12475, 13286, 17660, 19097, 16120], [11101, 11017, 13286, 14744, 17639, 15391]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 8]], 'representations': {'PC': {'code': 37564586580087815829051447178501, 'gens': [1, 2, 3, 5], 'pres': [6, 2, 2, 2, 3, 2, 3, 1982, 188, 50, 1155, 201, 2644, 1810, 916, 88, 293, 1739, 881]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [22581130, 7115160, 16858573]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 8101, 13286, 15473, 7433, 13933]}, 'Perm': {'d': 8, 'gens': [5329, 24, 1, 414, 11934, 12630]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3^2:C_2^2', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
