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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '8064.cv', 'ambient_counter': 74, 'ambient_order': 8064, 'ambient_tex': 'C_3:D_4\\times \\PGL(2,7)', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 6, 'counter': 256, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '8064.cv.112.p1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '112.p1.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': 112, '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': '72.46', 'subgroup_hash': 46, 'subgroup_order': 72, 'subgroup_tex': 'S_3\\times D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '8064.cv', 'aut_centralizer_order': None, 'aut_label': '112.p1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2016.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.e1.a1', '56.e1.a1', '56.i1.a1', '56.p1.a1'], 'contains': ['224.c1.a1', '224.j1.a1', '224.k1.a1', '224.z1.a1', '224.bc1.a1', '336.bd1.a1', '336.bn1.a1'], 'core': '1344.a1.a1', 'coset_action_label': None, 'count': 28, 'diagramx': [5679, -1, 2825, -1, 1683, -1, 3956, -1], 'generators': [127384253, 180707382159, 127370880, 87302054356, 174356582400], 'label': '8064.cv.112.p1.a1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.e1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '28.a1.a1', 'old_label': '112.p1.a1', 'projective_image': '4032.ds', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '112.p1.a1', 'subgroup_fusion': None, 'weyl_group': '72.46'}
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gps_groups • Show schema
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{'Agroup': True, '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, 3, 3], 'aut_gens': [[1, 2, 12], [37, 46, 12], [6, 25, 40], [1, 38, 12], [1, 10, 60], [37, 38, 12], [9, 2, 12], [1, 26, 12]], 'aut_group': '288.889', 'aut_hash': 889, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 10, 'aut_perms': [726, 374423, 1, 16687, 7, 126000, 782040], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 4, 1], [2, 9, 2, 1], [3, 2, 2, 1], [3, 4, 1, 1], [6, 2, 2, 1], [6, 4, 1, 1], [6, 6, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6\\wr C_2', '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': 12, 'autcentquo_group': '72.40', 'autcentquo_hash': 40, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 9, 2], [3, 2, 2], [3, 4, 1], [6, 2, 2], [6, 4, 1], [6, 6, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '36.10', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 46, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 9, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 336, 'exponent': 6, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '72.46', 'hash': 46, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 3], 'inner_gens': [[1, 10, 12], [5, 2, 60], [1, 26, 12]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'S_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 8], [4, 2]], 'label': '72.46', 'linC_count': 13, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 13, 'linQ_dim': 4, 'linQ_dim_count': 13, 'linR_count': 13, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*D6', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 6, 'number_conjugacy_classes': 18, 'number_divisions': 18, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 69, 'number_subgroups': 206, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 31], [3, 8], [6, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 38, 12], [42, 49, 40]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [6, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 3, '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, 8], [4, 2]], 'representations': {'PC': {'code': 1043840168412997, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -3, -2, -3, 101, 26, 122, 608, 58, 609]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [11780359, 26483311, 7115160]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8156, 8101, 13286, 13933]}, 'Perm': {'d': 8, 'gens': [25, 720, 24, 5760, 3]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times D_6', 'transitive_degree': 12, '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': 2, 'aut_exponent': 168, 'aut_gen_orders': [8, 6, 8, 24, 42, 2], 'aut_gens': [[127370880, 6230688345, 174396873823, 87302033280, 174356582400, 40279680], [127370880, 6230684590, 174443689675, 87302033280, 174356582400, 87091200], [127370880, 6350791542, 174396872310, 87181920000, 174356582400, 40279680], [127370880, 6350789730, 174443684931, 87181920000, 174356582400, 87091200], [127370880, 87181939220, 174443706996, 180707345280, 174356582400, 40279680], [127370880, 87181948662, 174443681520, 180707345280, 174356582400, 87091200], [127370880, 6230669411, 93445599600, 180707345280, 93405312000, 40279680]], 'aut_group': None, 'aut_hash': 6220676842330196211, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 64, 'aut_perms': [6765690151914089847212201295672638214748402091638040304938565862183151020830402270633546, 33614912120649885289221916154562738926950450816823641738777231271836029678989251121958783, 17318632292988974144219082233272293882471982651664470503762104119607554518221133360053881, 4949591828791616240915910258130989497985663352899696573126844466137494706207881158403415, 6569820225773679668057748803636846103662898517160469867549828909382007239263757025657962, 54311108272446078621795231091969906498822716056805333061355495731476192970144265082343333], 'aut_phi_ratio': 7.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 1], [2, 21, 1, 2], [2, 28, 2, 1], [2, 42, 1, 1], [2, 56, 1, 1], [2, 126, 1, 1], [2, 168, 1, 1], [3, 2, 1, 1], [3, 56, 1, 1], [3, 112, 1, 1], [4, 6, 1, 1], [4, 42, 1, 2], [4, 84, 1, 1], [4, 126, 1, 1], [4, 168, 1, 1], [4, 252, 1, 2], [6, 2, 1, 1], [6, 2, 2, 1], [6, 42, 1, 2], [6, 42, 2, 1], [6, 56, 1, 1], [6, 56, 2, 3], [6, 112, 1, 3], [6, 112, 2, 3], [6, 336, 1, 2], [7, 48, 1, 1], [8, 42, 2, 2], [8, 84, 1, 2], [8, 252, 1, 4], [12, 84, 1, 2], [12, 84, 2, 1], [12, 336, 1, 2], [14, 48, 1, 1], [14, 96, 1, 1], [14, 288, 1, 1], [21, 96, 1, 1], [24, 84, 2, 4], [28, 288, 1, 1], [42, 96, 1, 1], [42, 96, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times \\SO(3,7)\\times D_6', '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': 168, 'autcentquo_group': '2016.cw', 'autcentquo_hash': 117371359274444897, 'autcentquo_nilpotent': False, 'autcentquo_order': 2016, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times \\PGL(2,7)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 6, 1], [2, 21, 2], [2, 28, 2], [2, 42, 1], [2, 56, 1], [2, 126, 1], [2, 168, 1], [3, 2, 1], [3, 56, 1], [3, 112, 1], [4, 6, 1], [4, 42, 2], [4, 84, 1], [4, 126, 1], [4, 168, 1], [4, 252, 2], [6, 2, 3], [6, 42, 4], [6, 56, 7], [6, 112, 9], [6, 336, 2], [7, 48, 1], [8, 42, 4], [8, 84, 2], [8, 252, 4], [12, 84, 4], [12, 336, 2], [14, 48, 1], [14, 96, 1], [14, 288, 1], [21, 96, 1], [24, 84, 8], [28, 288, 1], [42, 96, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '4032.ds', 'commutator_count': 1, 'commutator_label': '1008.884', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '168.42'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 74, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['24.8', 1], ['336.208', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 1], [2, 21, 1, 2], [2, 28, 1, 2], [2, 42, 1, 1], [2, 56, 1, 1], [2, 126, 1, 1], [2, 168, 1, 1], [3, 2, 1, 1], [3, 56, 1, 1], [3, 112, 1, 1], [4, 6, 1, 1], [4, 42, 1, 2], [4, 84, 1, 1], [4, 126, 1, 1], [4, 168, 1, 1], [4, 252, 1, 2], [6, 2, 1, 1], [6, 2, 2, 1], [6, 42, 1, 2], [6, 42, 2, 1], [6, 56, 1, 5], [6, 56, 2, 1], [6, 112, 1, 5], [6, 112, 2, 2], [6, 336, 1, 2], [7, 48, 1, 1], [8, 42, 2, 2], [8, 84, 2, 1], [8, 252, 2, 2], [12, 84, 1, 2], [12, 84, 2, 1], [12, 336, 1, 2], [14, 48, 1, 1], [14, 96, 1, 1], [14, 288, 1, 1], [21, 96, 1, 1], [24, 84, 2, 2], [24, 84, 4, 1], [28, 288, 1, 1], [42, 96, 1, 1], [42, 96, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 9305856, 'exponent': 168, 'exponents_of_order': [7, 2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[12, 0, 6], [14, 0, 4], [16, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '4032.ds', 'hash': 1278949000154037978, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 168, 'inner_gen_orders': [1, 2, 6, 2, 3, 2], 'inner_gens': [[127370880, 6230688345, 174396873823, 87302033280, 174356582400, 40279680], [127370880, 6230688345, 93492414644, 180707345280, 93405312000, 87091200], [127370880, 87302038735, 174396873823, 180587232000, 174356582400, 40279680], [127370880, 180587270745, 93492414943, 87302033280, 93405312000, 87091200], [127370880, 87181958745, 174396873823, 180707345280, 174356582400, 40279680], [127370880, 6350801625, 174396873823, 87181920000, 174356582400, 40279680]], 'inner_hash': 5672968783997510635, 'inner_nilpotent': False, 'inner_order': 4032, 'inner_split': True, 'inner_tex': 'D_6\\times \\PGL(2,7)', 'inner_used': [2, 3, 6], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 10], [6, 12], [7, 8], [8, 8], [12, 15], [14, 10], [16, 10]], 'label': '8064.cv', 'linC_count': 48, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 40, 'linQ_dim': 10, 'linQ_dim_count': 40, 'linR_count': 120, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C3:D4*PGL(2,7)', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 64, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 81, 'number_divisions': 64, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 944, 'number_subgroup_classes': 1055, 'number_subgroups': 59522, 'old_label': None, 'order': 8064, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 499], [3, 170], [4, 972], [6, 2246], [7, 48], [8, 1344], [12, 1008], [14, 432], [21, 96], [24, 672], [28, 288], [42, 288]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[127370880, 6230688345, 174443685343, 87302033280, 174356582400, 40279680], [127370880, 180587270745, 93445603423, 87302033280, 93405312000, 40279680]], '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': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 2], [6, 4], [7, 8], [8, 8], [12, 7], [14, 6], [16, 6], [24, 4], [28, 2], [32, 2], [48, 1]], 'representations': {'Perm': {'d': 15, 'gens': [127370880, 6230688345, 174396873823, 87302033280, 174356582400, 40279680]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3:D_4\\times \\PGL(2,7)', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
