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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '32928.bb', 'ambient_counter': 28, 'ambient_order': 32928, 'ambient_tex': '(C_7\\times C_{14}^2):S_4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 1, 'counter': 150, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '32928.bb.168.bm1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '168.bm1.b1', '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': 168, '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': '196.8', 'subgroup_hash': 8, 'subgroup_order': 196, 'subgroup_tex': 'C_7^2:C_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '32928.bb', 'aut_centralizer_order': None, 'aut_label': '168.bm1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '16464.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['24.j1.b1', '84.k1.b1', '84.w1.b1', '84.x1.b1'], 'contains': ['336.w1.b1', '8232.i1.b1'], 'core': '32928.a1.a1', 'coset_action_label': None, 'count': 42, 'diagramx': [9604, -1, 5042, -1, 363, -1, 1672, -1], 'generators': [16465, 1368, 336, 31890], 'label': '32928.bb.168.bm1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '42.a1.a1', 'old_label': '168.bm1.b1', 'projective_image': '32928.bb', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '168.bm1.b1', 'subgroup_fusion': None, 'weyl_group': '392.37'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 336, 'aut_gen_orders': [2, 48, 7, 7], 'aut_gens': [[1, 4, 28], [3, 120, 144], [1, 84, 124], [105, 4, 28], [133, 4, 28]], 'aut_group': '4704.cb', 'aut_hash': 7849604463629877409, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4704, 'aut_permdeg': 49, 'aut_perms': [544781482416929792569308793075306315121615703308500576051340, 9400968260311734888963462477227695372390665472100964269889892, 544919034473404532370525612263695050106413560292402485259669025, 608241320315338468584064309132873996359892172666969923168664442], 'aut_phi_ratio': 56.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 49, 1, 1], [4, 49, 2, 1], [7, 4, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_{49}:C_2', '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': 336, 'autcentquo_group': '4704.cb', 'autcentquo_hash': 7849604463629877409, 'autcentquo_nilpotent': False, 'autcentquo_order': 4704, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_{49}:C_2', 'cc_stats': [[1, 1, 1], [2, 49, 1], [4, 49, 2], [7, 4, 12]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '196.8', 'commutator_count': 1, 'commutator_label': '49.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '7.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 49, 1, 1], [4, 49, 2, 1], [7, 4, 3, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 28, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 1, 12]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '196.8', 'hash': 8, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [4, 7, 7], 'inner_gens': [[1, 104, 68], [125, 4, 28], [185, 4, 28]], 'inner_hash': 8, 'inner_nilpotent': False, 'inner_order': 196, 'inner_split': False, 'inner_tex': 'C_7^2:C_4', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [4, 12]], 'label': '196.8', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 4, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C7^2:C4', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 16, 'number_divisions': 7, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 14, 'number_subgroups': 166, 'old_label': None, 'order': 196, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 49], [4, 98], [7, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 3], 'outer_gens': [[3, 120, 144], [1, 96, 12]], 'outer_group': '24.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [1560, 2164], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [12, 4]], 'representations': {'PC': {'code': 5071747118818355, 'gens': [1, 3, 4], 'pres': [4, -2, -2, -7, 7, 8, 1250, 150, 1091, 1351]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [13516898, 34720862, 473244, 4590980]}, 'Perm': {'d': 14, 'gens': [49816161360, 2566328399, 26983757673, 37362129120]}}, 'schur_multiplier': [7], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_7^2:C_4', 'transitive_degree': 14, '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': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 12, 21], 'aut_gens': [[1, 2, 6, 12, 168, 2352], [11153, 6092, 15234, 10878, 30576, 17736], [18101, 25070, 32178, 27678, 7056, 17760], [22027, 7204, 6468, 29718, 16512, 5880], [13863, 21494, 11172, 24618, 17760, 504]], 'aut_group': None, 'aut_hash': 3043542394710290185, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 395136, 'aut_permdeg': 84, 'aut_perms': [2439698442361621174409311127739328600799644967632827078250834369305222418965600223199914716556152977688141462947902468325939208, 2102478804359133820032948450409350747593429373654291365375627917534151895409569125788108758766159297400378241277798377078240578, 113317498029706250387221129687609238138162330532307563767384658693180370215076680591422578912787463239571381050866527400033088, 1925463173864060039804915942979624791214646940593347785081516485805592189417281994615617990314150143993151736581267567290027761], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 84, 1, 1], [2, 147, 2, 1], [2, 294, 1, 1], [3, 1568, 1, 1], [4, 84, 1, 1], [4, 4116, 2, 1], [7, 4, 6, 1], [7, 6, 3, 1], [7, 12, 3, 1], [7, 12, 6, 2], [7, 24, 2, 1], [7, 24, 3, 1], [14, 6, 3, 1], [14, 12, 3, 2], [14, 12, 6, 3], [14, 24, 3, 4], [14, 24, 6, 3], [14, 84, 6, 1], [14, 168, 3, 1], [14, 168, 6, 3], [14, 294, 6, 1], [14, 588, 3, 1], [21, 1568, 6, 1], [28, 84, 6, 1], [28, 168, 3, 1], [28, 168, 6, 3]], 'aut_supersolvable': False, 'aut_tex': 'C_7^3.C_2^4.C_6^2.C_2', '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': 84, 'autcentquo_group': None, 'autcentquo_hash': 3043542394710290185, 'autcentquo_nilpotent': False, 'autcentquo_order': 395136, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_7^3.C_2^4.C_6^2.C_2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 84, 1], [2, 147, 2], [2, 294, 1], [3, 1568, 1], [4, 84, 1], [4, 4116, 2], [7, 4, 6], [7, 6, 3], [7, 12, 15], [7, 24, 5], [14, 6, 3], [14, 12, 24], [14, 24, 30], [14, 84, 6], [14, 168, 21], [14, 294, 6], [14, 588, 3], [21, 1568, 6], [28, 84, 6], [28, 168, 21]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '32928.bb', 'commutator_count': 1, 'commutator_label': '16464.bw', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1', '7.1', '7.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 28, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 84, 1, 1], [2, 147, 1, 2], [2, 294, 1, 1], [3, 1568, 1, 1], [4, 84, 1, 1], [4, 4116, 1, 2], [7, 4, 6, 1], [7, 6, 3, 1], [7, 12, 3, 1], [7, 12, 6, 2], [7, 24, 2, 1], [7, 24, 3, 1], [14, 6, 3, 1], [14, 12, 3, 2], [14, 12, 6, 3], [14, 24, 3, 4], [14, 24, 6, 3], [14, 84, 6, 1], [14, 168, 3, 1], [14, 168, 6, 3], [14, 294, 3, 2], [14, 588, 3, 1], [21, 1568, 6, 1], [28, 84, 6, 1], [28, 168, 3, 1], [28, 168, 6, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 513, 'exponent': 84, 'exponents_of_order': [5, 3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[6, 0, 6], [6, 1, 6], [12, 0, 36], [12, 1, 12], [24, 0, 18], [24, 1, 12]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '32928.bb', 'hash': 471491061721889110, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [4, 21, 2, 14, 14, 14], 'inner_gens': [[1, 14218, 14532, 15846, 16488, 15288], [23897, 2, 14538, 14910, 7056, 17760], [14539, 14534, 6, 28380, 168, 30576], [20563, 19628, 4734, 12, 2184, 30576], [16777, 26042, 6, 348, 168, 2352], [22345, 20042, 4710, 4716, 168, 2352]], 'inner_hash': 471491061721889110, 'inner_nilpotent': False, 'inner_order': 32928, 'inner_split': True, 'inner_tex': '(C_7\\times C_{14}^2):S_4', 'inner_used': [1, 2, 5], 'irrC_degree': 6, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 6, 'irrep_stats': [[1, 2], [2, 1], [3, 6], [4, 12], [6, 25], [8, 6], [12, 78], [24, 35]], 'label': '32928.bb', 'linC_count': 12, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 2, 'linQ_dim': 18, 'linQ_dim_count': 2, 'linR_count': 6, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': '(C7*C14^2):S4', 'ngens': 9, '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': 41, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 165, 'number_divisions': 44, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 310, 'number_subgroup_classes': 392, 'number_subgroups': 65784, 'old_label': None, 'order': 32928, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 675], [3, 1568], [4, 8316], [7, 342], [14, 8586], [21, 9408], [28, 4032]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[22177, 16970, 21246, 2340, 2184, 30576], [23585, 6722, 15234, 29358, 2352, 17736]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [24, 724], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [3, 6], [6, 1], [18, 4], [24, 2], [36, 8], [48, 2], [72, 15], [144, 3]], 'representations': {'PC': {'code': '83547110175043380404356017695392123591341472552553809820104700156532998421049010235130774371583980888727672929475919637328424199670401563206025', 'gens': [1, 2, 3, 4, 6, 8], 'pres': [9, 2, 3, 2, 2, 7, 2, 7, 2, 7, 128250, 255925, 130474, 392366, 196274, 570459, 268392, 170301, 102, 45364, 30253, 1102, 890357, 190526, 9860, 158, 3030, 444543, 10617, 1100743, 639376, 366937, 183490, 214, 2286152, 2933, 381050, 190547]}, 'Perm': {'d': 25, 'gens': [54108240109785569772853, 646410290674537596302048]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_7\\times C_{14}^2):S_4', 'transitive_degree': 42, 'wreath_data': None, 'wreath_product': False}
