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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '3840.ft', 'ambient_counter': 150, 'ambient_order': 3840, 'ambient_tex': 'C_2\\times \\GL(2,\\mathbb{Z}/4):F_5', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 4, 'counter': 922, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '3840.ft.80.X', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '80.x1', 'outer_equivalence': True, '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': 80, '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': '48.43', 'subgroup_hash': 43, 'subgroup_order': 48, 'subgroup_tex': 'C_6:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3840.ft', 'aut_centralizer_order': 32, 'aut_label': '80.X', 'aut_quo_index': None, 'aut_stab_index': 80, 'aut_weyl_group': '24.14', 'aut_weyl_index': 2560, 'centralizer': '480.BY', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['16.BD', '20.O', '40.M', '40.BA'], 'contains': ['160.R', '160.T', '160.U', '160.V', '160.Y', '160.BD', '160.BE', '240.EP'], 'core': '960.B', 'coset_action_label': None, 'count': 80, 'diagramx': None, 'generators': [180625512383, 363000, 180623520000, 28385, 180623560014], 'label': '3840.ft.80.X', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.L', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '20.F', 'old_label': '80.x1', 'projective_image': '1920.240396', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '80.X', 'subgroup_fusion': None, 'weyl_group': '24.14'}
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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': 12, 'aut_gen_orders': [2, 2, 2, 2, 12], 'aut_gens': [[1, 2, 8], [25, 26, 8], [1, 2, 12], [1, 2, 40], [5, 6, 8], [25, 18, 12]], 'aut_group': '192.1147', 'aut_hash': 1147, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 11, 'aut_perms': [21144, 12501360, 8361601, 15301704, 19436403], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 6, 2, 1], [3, 2, 1, 1], [4, 6, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_6', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 6, 2], [3, 2, 1], [4, 6, 2], [6, 2, 7]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 43, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['24.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 6, 1, 2], [3, 2, 1, 1], [4, 6, 1, 2], [6, 2, 1, 3], [6, 2, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 168, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '24.14', 'hash': 43, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 3], 'inner_gens': [[1, 6, 8], [5, 2, 40], [1, 18, 8]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': False, 'inner_tex': 'D_6', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 10]], 'label': '48.43', 'linC_count': 16, 'linC_degree': 3, '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C6:D4', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 18, 'number_divisions': 16, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 54, 'number_subgroups': 108, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 19], [3, 2], [4, 12], [6, 14]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[25, 26, 8], [1, 2, 12], [5, 26, 8], [5, 2, 8]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [6, 289, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, '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]], 'representations': {'PC': {'code': 2906379043649, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -2, -2, -3, 61, 26, 408, 58, 409]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16563488, 35813463, 7233663]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [34720862, 6536214, 34603218, 16771677, 34409418]}, 'Perm': {'d': 9, 'gens': [5045, 120, 127, 16, 45360]}}, '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': 'C_6:D_4', 'transitive_degree': 24, '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': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [12, 4, 20, 4, 4, 4, 4], 'aut_gens': [[362886, 7187011321, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1], [5160, 262055416326, 19679017368, 368392147200, 367927, 348713532720, 368047, 363001, 99632695817, 363006], [126, 7186648327, 274467945305, 19678982400, 367927, 187768990086, 368047, 363001, 355898228704, 368040], [367921, 193558187527, 368392533430, 88176211200, 120, 187768632361, 368047, 363001, 99632730671, 368040], [367926, 355899818161, 274468308071, 180623520000, 120, 348713169966, 368047, 368040, 193556949535, 363006], [126, 262055784366, 274467951768, 368392147200, 367927, 93924593286, 368047, 368040, 7185426904, 363001], [362881, 262055421361, 88176239705, 368392147200, 5167, 280215936121, 5047, 5040, 355898580169, 6], [367921, 7186648327, 88176602471, 19678982400, 120, 348713169961, 368047, 363001, 262053826865, 368040]], 'aut_group': None, 'aut_hash': 1483974879331033022, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 61440, 'aut_permdeg': 48, 'aut_perms': [10055918268190300748547729577975912796417288146240168627905847, 9571827215033413321987878642561080868727378845530293091411501, 9571928231412665242382007266748958550722707603140080096325928, 10470344567759487031756284357298966805834717009831816552502962, 9573664062381657176684232901619813429037581969677903382944978, 4712618572699495250271753400423613303665718036802127174568110, 5622561253974010938964782955054597445118451544131763180202539], 'aut_phi_ratio': 60.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 5, 1, 2], [2, 5, 2, 1], [2, 6, 2, 1], [2, 10, 2, 1], [2, 12, 2, 1], [2, 15, 1, 2], [2, 15, 2, 1], [2, 30, 2, 1], [2, 60, 2, 1], [3, 8, 1, 1], [4, 10, 2, 4], [4, 12, 2, 3], [4, 30, 2, 4], [4, 30, 4, 4], [4, 60, 2, 7], [5, 4, 1, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 2, 1], [6, 40, 1, 2], [6, 40, 2, 1], [6, 80, 2, 1], [10, 4, 1, 1], [10, 4, 2, 1], [10, 8, 2, 1], [10, 12, 1, 2], [10, 12, 2, 1], [10, 24, 2, 1], [10, 48, 2, 1], [12, 80, 2, 4], [15, 32, 1, 1], [20, 48, 2, 3], [30, 32, 1, 1], [30, 32, 2, 1], [30, 32, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_5\\times A_4).C_2^4.C_2^6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '480.1189', 'autcentquo_hash': 1189, 'autcentquo_nilpotent': False, 'autcentquo_order': 480, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 3, 4], [2, 5, 4], [2, 6, 2], [2, 10, 2], [2, 12, 2], [2, 15, 4], [2, 30, 2], [2, 60, 2], [3, 8, 1], [4, 10, 8], [4, 12, 6], [4, 30, 24], [4, 60, 14], [5, 4, 1], [6, 8, 3], [6, 16, 2], [6, 40, 4], [6, 80, 2], [10, 4, 3], [10, 8, 2], [10, 12, 4], [10, 24, 2], [10, 48, 2], [12, 80, 8], [15, 32, 1], [20, 48, 6], [30, 32, 7]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '960.11361', 'commutator_count': 1, 'commutator_label': '120.43', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 150, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['1920.240035', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 3, 1, 4], [2, 5, 1, 4], [2, 6, 1, 2], [2, 10, 1, 2], [2, 12, 1, 2], [2, 15, 1, 4], [2, 30, 1, 2], [2, 60, 1, 2], [3, 8, 1, 1], [4, 10, 2, 4], [4, 12, 1, 6], [4, 30, 2, 12], [4, 60, 1, 6], [4, 60, 2, 4], [5, 4, 1, 1], [6, 8, 1, 3], [6, 16, 1, 2], [6, 40, 1, 4], [6, 80, 1, 2], [10, 4, 1, 3], [10, 8, 1, 2], [10, 12, 1, 4], [10, 24, 1, 2], [10, 48, 1, 2], [12, 80, 2, 4], [15, 32, 1, 1], [20, 48, 1, 6], [30, 32, 1, 3], [30, 32, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1023684480, 'exponent': 60, 'exponents_of_order': [8, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '1920.240396', 'hash': 7250606812080537101, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 4, 6, 2, 2, 10, 1, 2, 4, 2], 'inner_gens': [[362886, 7186648321, 274467951768, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1], [126, 7187011321, 88176613974, 19678982400, 367927, 348713527686, 5047, 5040, 262054189865, 1], [367920, 193558187526, 274467951775, 88176211200, 5167, 280215941160, 5047, 1, 262054155022, 6], [362886, 7187011321, 180623554975, 19678982400, 5167, 280215936126, 5047, 5040, 99632735704, 1], [362886, 7186648321, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1], [362886, 262055416321, 180623554968, 368392147200, 5167, 187768627326, 5047, 5040, 193556966104, 1], [362886, 7187011321, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1], [362886, 7187011321, 274467956809, 19678982400, 5167, 187768627326, 5047, 5040, 355898586665, 1], [362886, 99634325160, 368392175705, 274467916800, 5167, 348713164926, 5047, 1, 355898591704, 5040], [362886, 7187011321, 274467956814, 19678982400, 5167, 187768627326, 5047, 5040, 355898586665, 1]], 'inner_hash': 11361, 'inner_nilpotent': False, 'inner_order': 960, 'inner_split': False, 'inner_tex': 'C_2\\times F_5\\times S_4', 'inner_used': [1, 2, 3, 9], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [3, 32], [4, 12], [6, 8], [8, 10], [12, 8], [24, 2]], 'label': '3840.ft', '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': None, 'name': 'C2*GL(2,Z/4):F5', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 68, 'number_characteristic_subgroups': 76, 'number_conjugacy_classes': 128, 'number_divisions': 102, 'number_normal_subgroups': 248, 'number_subgroup_autclasses': 2198, 'number_subgroup_classes': 5338, 'number_subgroups': 77316, 'old_label': None, 'order': 3840, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 335], [3, 8], [4, 1712], [5, 4], [6, 376], [10, 220], [12, 640], [15, 32], [20, 288], [30, 224]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 19678982400, 0], 'outer_gens': [[367921, 7187016366, 274467956808, 19678982400, 120, 187768632361, 5047, 5040, 355898591704, 1], [362886, 7187016366, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898586671, 1], [362886, 7187016366, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1], [362886, 7187011321, 368392182175, 19678982400, 5167, 93924230526, 5047, 5040, 193556966104, 1], [126, 7186653366, 274467951775, 19678982400, 367927, 187768990086, 368047, 368040, 355898586671, 363001]], 'outer_group': '64.202', 'outer_hash': 202, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [1174447, 368063, 368040, 367920, 363022], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3:D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [3, 16], [4, 18], [6, 12], [8, 6], [12, 10], [16, 2], [24, 2]], 'representations': {'PC': {'code': '339095252087333614047069559593955708549354442679144545081731099001340733320034667623523907108046481807596665612617646192', 'gens': [1, 2, 4, 6, 9, 10], 'pres': [10, -2, -2, -2, -2, -2, -2, -3, -5, -2, 2, 42681, 51, 50093, 7783, 113, 19614, 5024, 59525, 64335, 9145, 11315, 5925, 175, 206086, 49296, 4506, 17956, 13766, 276, 15377, 15387, 3877, 3887, 32438, 5458, 4118, 9059, 1569]}, 'Perm': {'d': 15, 'gens': [362886, 7187011321, 274467951775, 19678982400, 5167, 187768627326, 5047, 5040, 355898591704, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times \\GL(2,\\mathbb{Z}/4):F_5', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
