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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '6531840.b', 'ambient_counter': 2, 'ambient_order': 6531840, 'ambient_tex': '\\PSOMinus(6,3)', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 868, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6531840.b.408240.bt1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '408240.bt1.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': 408240, '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': '16.11', 'subgroup_hash': 11, 'subgroup_order': 16, 'subgroup_tex': 'C_2\\times D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6531840.b', 'aut_centralizer_order': None, 'aut_label': '408240.bt1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '816480.s1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['204120.p1.a1', '204120.u1.a1', '204120.bk1.a1', '204120.bl1.a1', '204120.bl1.a2', '204120.bn1.a1', '204120.bn1.a2', '204120.bo2.a1', '204120.bo2.a2', '204120.bp1.a1', '204120.bq1.a1', '204120.br1.a1', '204120.bt1.a1', '204120.bt1.a2', '204120.bu1.a1'], 'contains': ['816480.t1.a1', '816480.w1.a1', '816480.y1.a1', '816480.y1.a2'], 'core': '6531840.a1.a1', 'coset_action_label': None, 'count': 25515, 'diagramx': None, 'generators': [78284631970101376, 78313591464846418, 31421448411707001, 109177104520513901], 'label': '6531840.b.408240.bt1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '25515.a1.a1', 'old_label': '408240.bt1.a1', 'projective_image': '6531840.b', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '408240.bt1.a1', 'subgroup_fusion': None, 'weyl_group': '32.27'}
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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], 'aut_gens': [[126, 55, 289, 288], [127, 55, 289, 288], [126, 265, 1, 288], [54, 127, 289, 288], [414, 55, 289, 288], [127, 54, 289, 288], [414, 265, 289, 288]], 'aut_group': '64.138', 'aut_hash': 138, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 8, 'aut_perms': [2309, 526, 5329, 3043, 12316, 18498], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [4, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr 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, 3], [2, 2, 4], [4, 2, 2]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [4, 2, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 21, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 1], 'inner_gens': [[126, 265, 289, 288], [414, 55, 289, 288], [126, 55, 289, 288], [126, 55, 289, 288]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.11', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 8, 'linQ_dim': 3, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 10, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 27, 'number_subgroups': 35, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 11], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 151, 0], 'outer_gens': [[54, 127, 289, 288], [127, 55, 289, 288], [415, 54, 1, 288], [127, 54, 289, 288]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', '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], [2, 2]], 'representations': {'PC': {'code': 8772, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 78, 34]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16322, 16432, 3198]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8912, 8156, 13286, 14044]}, 'Perm': {'d': 6, 'gens': [126, 55, 289, 288]}}, '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': 'C_2\\times D_4', 'transitive_degree': 8, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 2520, 'aut_gen_orders': [8, 9, 2], 'aut_gens': [[50077316156099761, 16677764777968770, 50077309149050176], [116733516203203951, 86554451355277937, 50834220306819262], [39684763801295041, 102514216690548264, 54062120658997408], [66407004122010619, 67764971263556240, 51385242658610947]], 'aut_group': '26127360.d', 'aut_hash': 2943447302241579943, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 26127360, 'aut_permdeg': 540, 'aut_perms': 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'aut_phi_ratio': 17.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 540, 1, 1], [2, 2835, 1, 1], [2, 4536, 1, 1], [3, 560, 1, 1], [3, 3360, 2, 1], [3, 40320, 1, 1], [4, 5670, 1, 1], [4, 34020, 1, 1], [4, 51030, 1, 1], [4, 204120, 1, 1], [5, 653184, 1, 1], [6, 30240, 1, 1], [6, 45360, 1, 1], [6, 90720, 2, 1], [6, 181440, 2, 1], [6, 362880, 1, 1], [7, 466560, 2, 1], [8, 408240, 1, 2], [9, 241920, 2, 1], [10, 653184, 1, 1], [12, 45360, 2, 1], [12, 181440, 2, 1], [12, 272160, 1, 1], [14, 466560, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PGammaU(4,3)', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2520, 'autcentquo_group': '26127360.d', 'autcentquo_hash': 2943447302241579943, 'autcentquo_nilpotent': False, 'autcentquo_order': 26127360, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PGammaU(4,3)', 'cc_stats': [[1, 1, 1], [2, 540, 1], [2, 2835, 1], [2, 4536, 1], [3, 560, 1], [3, 3360, 2], [3, 40320, 1], [4, 5670, 1], [4, 34020, 1], [4, 51030, 1], [4, 204120, 1], [5, 653184, 1], [6, 30240, 1], [6, 45360, 1], [6, 90720, 2], [6, 181440, 2], [6, 362880, 1], [7, 466560, 2], [8, 408240, 2], [9, 241920, 2], [10, 653184, 1], [12, 45360, 2], [12, 181440, 2], [12, 272160, 1], [14, 466560, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6531840.b', 'commutator_count': 1, 'commutator_label': '3265920.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3265920.a'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 540, 1, 1], [2, 2835, 1, 1], [2, 4536, 1, 1], [3, 560, 1, 1], [3, 3360, 1, 2], [3, 40320, 1, 1], [4, 5670, 1, 1], [4, 34020, 1, 1], [4, 51030, 1, 1], [4, 204120, 1, 1], [5, 653184, 1, 1], [6, 30240, 1, 1], [6, 45360, 1, 1], [6, 90720, 1, 2], [6, 181440, 1, 2], [6, 362880, 1, 1], [7, 466560, 2, 1], [8, 408240, 1, 2], [9, 241920, 1, 2], [10, 653184, 1, 1], [12, 45360, 2, 1], [12, 181440, 1, 2], [12, 272160, 1, 1], [14, 466560, 2, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': None, 'exponent': 2520, 'exponents_of_order': [8, 6, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3, 5, 7], 'faithful_reps': [[21, 1, 2], [35, 1, 4], [90, 1, 2], [140, 1, 2], [189, 1, 2], [210, 1, 2], [315, 1, 4], [420, 1, 2], [560, 0, 2], [560, 1, 2], [640, 0, 4], [729, 1, 2], [896, 1, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6531840.b', 'hash': 8109001797732275, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 2520, 'inner_gen_orders': [4, 9, 2], 'inner_gens': [[50077316156099761, 16677764390029962, 50077309149050176], [113263024647040643, 16677764777968770, 102350780684006294], [50077316156099761, 16677771359865555, 50077309149050176]], 'inner_hash': 8109001797732275, 'inner_nilpotent': False, 'inner_order': 6531840, 'inner_split': False, 'inner_tex': '\\PSOMinus(6,3)', 'inner_used': [1, 2, 3], 'irrC_degree': 21, 'irrQ_degree': 21, 'irrQ_dim': 21, 'irrR_degree': 21, 'irrep_stats': [[1, 2], [21, 2], [35, 4], [90, 2], [140, 2], [189, 2], [210, 2], [315, 4], [420, 2], [560, 4], [640, 4], [729, 2], [896, 2]], 'label': '6531840.b', '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': False, 'name': 'PSO-(6,3)', 'ngens': 3, '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, 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, 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, 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, 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, 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, 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, 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, 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': 26, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 34, 'number_divisions': 31, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 659, 'number_subgroup_classes': 983, 'number_subgroups': 33884546, 'old_label': None, 'order': 6531840, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 7911], [3, 47600], [4, 294840], [5, 653184], [6, 982800], [7, 933120], [8, 816480], [9, 483840], [10, 653184], [12, 725760], [14, 933120]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [111561641817188651, 85135733989493878], 'outer_gens': [[39735820382953446, 71090868607080088, 29205174101958690], [91077095702793019, 63514781037897050, 81000600703675771]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 112, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [21, 2], [35, 4], [90, 2], [140, 2], [189, 2], [210, 2], [315, 4], [420, 2], [560, 2], [729, 2], [896, 2], [1120, 1], [1280, 2]], 'representations': {'Lie': [{'d': 6, 'q': 3, 'gens': [50077309149050176, 16677764777968770, 50077316156099761], 'family': 'PSOMinus'}], 'Perm': {'d': 112, 'gens': [6499473806904459303242362343133880436412780409967094809599787282508804516736980463479599825181504605491761394995722278924463064589108210295655339707058427338075933357404, 1795010208112141724888994965365846591139959375438226956893227218149460206340525503292272660592395014476842888819448440958943129420920810140497154716204515290809202724237688529775646, 2661782454126690432092439613961981365725368869407251260394735671343513526558263396044327423772229610216110169054530834754280039897094016770932398773495750018022219292351911425]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\PSOMinus(6,3)', 'transitive_degree': 112, 'wreath_data': None, 'wreath_product': False}
