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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '48000.ba', 'ambient_counter': 27, 'ambient_order': 48000, 'ambient_tex': 'D_5^2.C_2^2\\times S_5', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 24000, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '48000.ba.2.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': False, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '24000.bk', 'subgroup_hash': 3577048968767335338, 'subgroup_order': 24000, 'subgroup_tex': 'D_5:F_5\\times S_5', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '48000.ba', 'aut_centralizer_order': None, 'aut_label': '2.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '48000.a1', 'complements': ['24000.f1', '24000.b1', '24000.e1'], 'conjugacy_class_count': 3, 'contained_in': ['1.a1'], 'contains': ['4.a1', '4.b1', '4.c1', '4.e1', '4.f1', '4.h1', '4.i1', '10.b1', '10.e1', '12.a1', '20.i1'], 'core': '2.a1', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [67, 52, 87179460524, 449377, 857520, 1219747, 367913912519], 'label': '48000.ba.2.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1', 'normal_contained_in': ['1.a1'], 'normal_contains': ['4.a1', '4.b1', '4.c1', '4.e1', '4.f1', '4.h1', '4.i1'], 'normalizer': '1.a1', 'old_label': '2.a1', 'projective_image': '48000.ba', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.a1', 'subgroup_fusion': None, 'weyl_group': '48000.ba'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [2, 3, 4, 2, 4, 2, 5, 4, 5], 'aut_gens': [[87179829127, 2771904, 367395765127, 93, 46177, 1219747, 857520], [41454727, 2771904, 187810081927, 93, 46177, 1219747, 857520], [180585141127, 2771904, 106858811527, 93, 46177, 1219747, 857520], [87179829239, 2902276, 367395765239, 413280, 46177, 90746, 37], [87179058082, 2771904, 367394994082, 413280, 1456644, 1219747, 52], [87178659146, 2050553, 280256220746, 1089397, 46177, 766826, 413317], [87178381987, 3578393, 367394317987, 1089360, 484577, 1538002, 37], [87178381987, 2050531, 367394317987, 93, 1139804, 368039, 857520], [87179829127, 2499013, 367395765127, 857632, 46177, 367927, 1573312], [87179510999, 2050553, 367395446999, 93, 811497, 90802, 857520]], 'aut_group': '96000.p', 'aut_hash': 4764451904582591359, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96000, 'aut_permdeg': 45, 'aut_perms': [2478465081093365104185077352816320237266210540579864614, 1874312450459296373313066493739839868432348914912660128, 16013445943065108483846227237487911233159217564513289496, 4547913042473204462462022773778298519406245004445592, 87784074797889226501355786778527087988411430840760613731, 29315619148376827348248490970447065363424178577318391273, 53247786204403785886404409935879746210353479874088519657, 114371714779761276954507875699081777075314818161524024568, 32013065378059326591012596184388348621972893074484837472], 'aut_phi_ratio': 15.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 5, 2, 1], [2, 10, 1, 1], [2, 15, 1, 1], [2, 25, 1, 1], [2, 50, 2, 1], [2, 75, 2, 1], [2, 250, 1, 1], [2, 375, 1, 1], [3, 20, 1, 1], [4, 25, 1, 2], [4, 25, 2, 1], [4, 30, 1, 1], [4, 150, 2, 1], [4, 250, 1, 2], [4, 250, 2, 1], [4, 375, 1, 2], [4, 375, 2, 1], [4, 750, 1, 3], [4, 750, 2, 1], [5, 4, 2, 1], [5, 8, 2, 1], [5, 24, 1, 1], [5, 96, 2, 1], [5, 192, 2, 1], [6, 20, 1, 1], [6, 100, 2, 2], [6, 500, 1, 2], [10, 20, 2, 1], [10, 40, 2, 1], [10, 60, 2, 1], [10, 80, 2, 1], [10, 120, 2, 2], [10, 200, 2, 1], [10, 300, 2, 1], [10, 480, 2, 1], [10, 600, 1, 1], [12, 500, 1, 4], [12, 500, 2, 2], [15, 80, 2, 1], [15, 160, 2, 1], [20, 120, 2, 1], [20, 240, 2, 1], [20, 600, 1, 2], [20, 600, 2, 2], [30, 80, 2, 1], [30, 160, 2, 1], [30, 400, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\wr C_2\\times S_5', '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': 120, 'autcentquo_group': '96000.p', 'autcentquo_hash': 4764451904582591359, 'autcentquo_nilpotent': False, 'autcentquo_order': 96000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\wr C_2\\times S_5', 'cc_stats': [[1, 1, 1], [2, 5, 2], [2, 10, 1], [2, 15, 1], [2, 25, 1], [2, 50, 2], [2, 75, 2], [2, 250, 1], [2, 375, 1], [3, 20, 1], [4, 25, 4], [4, 30, 1], [4, 150, 2], [4, 250, 4], [4, 375, 4], [4, 750, 5], [5, 4, 2], [5, 8, 2], [5, 24, 1], [5, 96, 2], [5, 192, 2], [6, 20, 1], [6, 100, 4], [6, 500, 2], [10, 20, 2], [10, 40, 2], [10, 60, 2], [10, 80, 2], [10, 120, 4], [10, 200, 2], [10, 300, 2], [10, 480, 2], [10, 600, 1], [12, 500, 8], [15, 80, 2], [15, 160, 2], [20, 120, 2], [20, 240, 2], [20, 600, 6], [30, 80, 2], [30, 160, 2], [30, 400, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '24000.bk', 'commutator_count': 1, 'commutator_label': '1500.112', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '5.1', '60.5'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 37, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['120.34', 1], ['200.42', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 2], [2, 10, 1, 1], [2, 15, 1, 1], [2, 25, 1, 1], [2, 50, 1, 2], [2, 75, 1, 2], [2, 250, 1, 1], [2, 375, 1, 1], [3, 20, 1, 1], [4, 25, 2, 2], [4, 30, 1, 1], [4, 150, 1, 2], [4, 250, 2, 2], [4, 375, 2, 2], [4, 750, 1, 1], [4, 750, 2, 2], [5, 4, 1, 2], [5, 8, 1, 2], [5, 24, 1, 1], [5, 96, 1, 2], [5, 192, 1, 2], [6, 20, 1, 1], [6, 100, 1, 4], [6, 500, 1, 2], [10, 20, 1, 2], [10, 40, 1, 2], [10, 60, 1, 2], [10, 80, 1, 2], [10, 120, 1, 4], [10, 200, 1, 2], [10, 300, 1, 2], [10, 480, 1, 2], [10, 600, 1, 1], [12, 500, 2, 4], [15, 80, 1, 2], [15, 160, 1, 2], [20, 120, 1, 2], [20, 240, 1, 2], [20, 600, 1, 2], [20, 600, 2, 2], [30, 80, 1, 2], [30, 160, 1, 2], [30, 400, 1, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 40352256, 'exponent': 60, 'exponents_of_order': [6, 3, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[32, 1, 4], [40, 1, 4], [48, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '24000.bk', 'hash': 3577048968767335338, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 4, 6, 5, 4, 2, 5], 'inner_gens': [[87179829127, 3578338, 280696475527, 112, 811524, 766826, 1573200], [87179058026, 2771904, 367394994026, 1573200, 484604, 1538002, 93], [41454727, 3578338, 367395765127, 112, 811524, 766826, 1573200], [87179829187, 2499113, 367395765187, 93, 46124, 1219799, 857520], [87178659146, 2499013, 367394595146, 52, 46177, 1537987, 1089360], [87178382039, 2499058, 367394318039, 112, 484657, 1219747, 1573200], [87179058007, 3265913, 367394994007, 93, 1456657, 1537987, 857520]], 'inner_hash': 3577048968767335338, 'inner_nilpotent': False, 'inner_order': 24000, 'inner_split': False, 'inner_tex': 'D_5:F_5\\times S_5', 'inner_used': [1, 2, 3, 5], 'irrC_degree': 32, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 32, 'irrep_stats': [[1, 16], [4, 24], [5, 16], [6, 8], [8, 4], [16, 8], [20, 8], [24, 4], [32, 4], [40, 4], [48, 2]], 'label': '24000.bk', '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': 'D5:F5*S5', '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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 63, 'number_characteristic_subgroups': 28, 'number_conjugacy_classes': 98, 'number_divisions': 84, 'number_normal_subgroups': 52, 'number_subgroup_autclasses': 1199, 'number_subgroup_classes': 1860, 'number_subgroups': 166180, 'old_label': None, 'order': 24000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 935], [3, 20], [4, 6680], [5, 624], [6, 1420], [10, 3440], [12, 4000], [15, 480], [20, 4320], [30, 2080]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [736582, 3219806], 'outer_gens': [[87179829127, 3175156, 367395765127, 1089397, 46164, 1537987, 1089412], [87179829127, 2499013, 367395765127, 857632, 46177, 367927, 1573312]], '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, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 16], [5, 8], [6, 4], [8, 8], [10, 4], [12, 2], [16, 8], [20, 8], [24, 4], [32, 4], [40, 4], [48, 2]], 'representations': {'Perm': {'d': 15, 'gens': [87179829127, 2771904, 367395765127, 93, 46177, 1219747, 857520]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_5:F_5\\times S_5', 'transitive_degree': 50, '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': 5, 'aut_exponent': 120, 'aut_gen_orders': [20, 12, 12], 'aut_gens': [[449377, 3542300, 367395396524, 1219747, 93, 87179460524, 857520, 1537920], [1139822, 2454380, 186851025395, 766826, 1573200, 268241380595, 112, 82], [2408282, 1456662, 349713180602, 367946, 1089472, 6229116602, 413392, 3507041], [449297, 2095876, 367913266644, 1219687, 112, 40037844, 413280, 367920]], 'aut_group': None, 'aut_hash': 6521009520396367745, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288000, 'aut_permdeg': 40, 'aut_perms': [171679344749326456735955974535182512942098611422, 279392061307042064185691873793380510757353270028, 664780456146965604048964999409731470729906989469], 'aut_phi_ratio': 22.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 10, 1, 1], [2, 10, 3, 1], [2, 15, 1, 1], [2, 25, 1, 1], [2, 100, 3, 1], [2, 150, 3, 1], [2, 250, 1, 1], [2, 375, 1, 1], [3, 20, 1, 1], [4, 25, 1, 2], [4, 30, 1, 1], [4, 50, 3, 1], [4, 250, 1, 2], [4, 300, 3, 1], [4, 375, 1, 2], [4, 500, 3, 1], [4, 750, 1, 3], [4, 750, 3, 1], [4, 1500, 3, 1], [5, 8, 3, 1], [5, 24, 1, 1], [5, 192, 3, 1], [6, 20, 1, 1], [6, 200, 3, 2], [6, 500, 1, 2], [10, 40, 3, 1], [10, 80, 3, 1], [10, 120, 3, 1], [10, 240, 3, 1], [10, 400, 3, 1], [10, 600, 1, 1], [10, 600, 3, 1], [10, 960, 3, 1], [12, 500, 1, 4], [12, 1000, 3, 2], [15, 160, 3, 1], [20, 240, 3, 1], [20, 600, 1, 2], [20, 1200, 3, 2], [30, 160, 3, 1], [30, 800, 3, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_5:D_5.A_4.C_4.S_5', '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': 120, 'autcentquo_group': None, 'autcentquo_hash': 6521009520396367745, 'autcentquo_nilpotent': False, 'autcentquo_order': 288000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5:D_5.A_4.C_4.S_5', 'cc_stats': [[1, 1, 1], [2, 10, 4], [2, 15, 1], [2, 25, 1], [2, 100, 3], [2, 150, 3], [2, 250, 1], [2, 375, 1], [3, 20, 1], [4, 25, 2], [4, 30, 1], [4, 50, 3], [4, 250, 2], [4, 300, 3], [4, 375, 2], [4, 500, 3], [4, 750, 6], [4, 1500, 3], [5, 8, 3], [5, 24, 1], [5, 192, 3], [6, 20, 1], [6, 200, 6], [6, 500, 2], [10, 40, 3], [10, 80, 3], [10, 120, 3], [10, 240, 3], [10, 400, 3], [10, 600, 4], [10, 960, 3], [12, 500, 4], [12, 1000, 6], [15, 160, 3], [20, 240, 3], [20, 600, 2], [20, 1200, 6], [30, 160, 3], [30, 800, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '48000.ba', 'commutator_count': 1, 'commutator_label': '3000.ch', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1', '60.5'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 27, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['120.34', 1], ['400.207', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 10, 1, 4], [2, 15, 1, 1], [2, 25, 1, 1], [2, 100, 1, 3], [2, 150, 1, 3], [2, 250, 1, 1], [2, 375, 1, 1], [3, 20, 1, 1], [4, 25, 2, 1], [4, 30, 1, 1], [4, 50, 1, 3], [4, 250, 2, 1], [4, 300, 1, 3], [4, 375, 2, 1], [4, 500, 1, 3], [4, 750, 1, 4], [4, 750, 2, 1], [4, 1500, 1, 3], [5, 8, 1, 3], [5, 24, 1, 1], [5, 192, 1, 3], [6, 20, 1, 1], [6, 200, 1, 6], [6, 500, 1, 2], [10, 40, 1, 3], [10, 80, 1, 3], [10, 120, 1, 3], [10, 240, 1, 3], [10, 400, 1, 3], [10, 600, 1, 4], [10, 960, 1, 3], [12, 500, 2, 2], [12, 1000, 1, 6], [15, 160, 1, 3], [20, 240, 1, 3], [20, 600, 2, 1], [20, 1200, 1, 6], [30, 160, 1, 3], [30, 800, 1, 6]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 5593464495360, 'exponent': 60, 'exponents_of_order': [7, 3, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[32, 1, 12], [40, 1, 12], [48, 1, 6]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '48000.ba', 'hash': 7364175325132069205, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [4, 4, 12, 2, 5, 4, 5, 2], 'inner_gens': [[449377, 3219731, 367394348217, 1219747, 52, 87178412217, 413280, 367920], [1139835, 3542300, 367395683795, 766919, 413280, 87179747795, 37, 7], [120977, 2822340, 367395396524, 90727, 52, 41086124, 413280, 766800], [449377, 2851851, 367395729137, 1219747, 112, 87179793137, 1573200, 90720], [449324, 2167166, 367395396537, 1219799, 93, 87179460537, 857520, 1537920], [120977, 2822340, 280696106924, 90727, 52, 87179460524, 413280, 766800], [736657, 3144891, 367394348204, 1537987, 93, 87178412204, 857520, 766800], [1502017, 3219816, 367394348204, 766867, 93, 87178412204, 1573200, 1537920]], 'inner_hash': 7364175325132069205, 'inner_nilpotent': False, 'inner_order': 48000, 'inner_split': False, 'inner_tex': 'D_5^2.C_2^2\\times S_5', 'inner_used': [1, 2, 3, 6, 8], 'irrC_degree': 32, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 32, 'irrep_stats': [[1, 16], [2, 4], [4, 16], [5, 16], [6, 8], [8, 16], [10, 4], [12, 2], [32, 12], [40, 12], [48, 6]], 'label': '48000.ba', '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': 'D5^2.C2^2*S5', '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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 56, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 112, 'number_divisions': 105, 'number_normal_subgroups': 90, 'number_subgroup_autclasses': 1523, 'number_subgroup_classes': 4169, 'number_subgroups': 605610, 'old_label': None, 'order': 48000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1455], [3, 20], [4, 12880], [5, 624], [6, 2220], [10, 7920], [12, 8000], [15, 480], [20, 9120], [30, 5280]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [119, 121035], 'outer_gens': [[449297, 3265824, 367395396524, 1219687, 52, 87179460524, 857520, 1537920], [2851820, 2771993, 367397769531, 766867, 413332, 87181833531, 857613, 3175138]], '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': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [4, 18], [5, 16], [6, 8], [8, 12], [16, 2], [20, 2], [24, 1], [32, 12], [40, 12], [48, 6]], 'representations': {'Perm': {'d': 15, 'gens': [449377, 3542300, 367395396524, 1219747, 93, 87179460524, 857520, 1537920]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_5^2.C_2^2\\times S_5', 'transitive_degree': 50, 'wreath_data': None, 'wreath_product': False}
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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}
