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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '768.1090235', 'ambient_counter': 1090235, 'ambient_order': 768, 'ambient_tex': 'C_2^5:S_4', 'central': False, 'central_factor': False, 'centralizer_order': 128, 'characteristic': False, 'core_order': 4, 'counter': 27, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '768.1090235.12.l1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '12.l1', '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': 12, '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': '64.267', 'subgroup_hash': 267, 'subgroup_order': 64, 'subgroup_tex': 'C_2^6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '768.1090235', 'aut_centralizer_order': 128, 'aut_label': '12.l1', 'aut_quo_index': None, 'aut_stab_index': 42, 'aut_weyl_group': '24.12', 'aut_weyl_index': 5376, 'centralizer': '6.a1', 'complements': None, 'conjugacy_class_count': 7, 'contained_in': ['6.a1'], 'contains': ['24.s1', '24.s2', '24.w1', '24.w2', '24.x1', '24.y1', '24.y2', '24.be1'], 'core': '192.a1', 'coset_action_label': None, 'count': 42, 'diagramx': [5395, -1, 2690, -1], 'generators': [21196264327, 27430179961, 2438997120, 20717187120, 1959920047, 14856750720], 'label': '768.1090235.12.l1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '6.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.a1', 'old_label': '12.l1', 'projective_image': '768.1090235', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.l1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '64.267', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 78120, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2, 4, 8, 16, 32], [13, 2, 4, 8, 16, 32], [44, 57, 28, 55, 38, 54]], 'aut_group': '20158709760.a', 'aut_hash': 1127810123506825908, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20158709760, 'aut_permdeg': 63, 'aut_perms': [1088780861340731732174280893943159511512640016119869733192292622602376850177766021539680, 193386930810790965654717163452553945898231780614988004231894671695879290786822421539277], 'aut_phi_ratio': 629959680.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 63, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(6,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 78120, 'autcent_group': '20158709760.a', 'autcent_hash': 1127810123506825908, 'autcent_nilpotent': False, 'autcent_order': 20158709760, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(6,2)', '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, 63]], 'center_label': '64.267', 'center_order': 64, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 267, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 6]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 63]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3, 5, 7, 31], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '64.267', 'hash': 267, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8, 16, 32], [1, 2, 4, 8, 16, 32], [1, 2, 4, 8, 16, 32], [1, 2, 4, 8, 16, 32], [1, 2, 4, 8, 16, 32], [1, 2, 4, 8, 16, 32]], '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, 64]], 'label': '64.267', 'linC_count': None, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^6', 'ngens': 6, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 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': 64, 'number_divisions': 64, 'number_normal_subgroups': 2825, 'number_subgroup_autclasses': 7, 'number_subgroup_classes': 2825, 'number_subgroups': 2825, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 63]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 78120, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[13, 2, 4, 8, 16, 32], [44, 57, 28, 55, 38, 54]], 'outer_group': '20158709760.a', 'outer_hash': 1127810123506825908, 'outer_nilpotent': False, 'outer_order': 20158709760, 'outer_permdeg': 63, 'outer_perms': [1088780861340731732174280893943159511512640016119869733192292622602376850177766021539680, 193386930810790965654717163452553945898231780614988004231894671695879290786822421539277], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(6,2)', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 6, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 64]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4, 5, 6], 'pres': [6, -2, 2, 2, 2, 2, 2]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101750005088214, 125101750005092586, 25038659807093174, 125101729084386182, 24992906222183252, 125101729074820244]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [5226561, 7292993, 5128257, 20225089, 19896385, 6210625]}, 'Perm': {'d': 12, 'gens': [39916800, 362880, 5040, 120, 6, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^6', 'transitive_degree': 64, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [42, 14], 'aut_gens': [[6442294336, 127, 43591264, 14377673646, 21196264327, 27430179840, 15335757360, 21675270960, 26951102640], [32776415878, 127, 13016551385, 21196264320, 27909186480, 7, 121, 15335757367, 20717187241], [6362380369, 127, 21163207662, 26951102647, 21675271086, 15335757367, 27430179966, 7, 126]], 'aut_group': '129024.j', 'aut_hash': 1511937657504715492, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 129024, 'aut_permdeg': 48, 'aut_perms': [11697565001173319743004682719966316637398311868799921724876348, 9625503872006983775618330163177500286967613279253755025993968], 'aut_phi_ratio': 504.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 7, 2], [2, 6, 7, 2], [2, 24, 2, 1], [3, 128, 1, 1], [4, 24, 14, 1], [6, 128, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^7.(S_3\\times \\GL(3,2))', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 168, 'autcentquo_group': '64512.b', 'autcentquo_hash': 3460061284310155072, 'autcentquo_nilpotent': False, 'autcentquo_order': 64512, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^6:(S_3\\times \\GL(3,2))', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 14], [2, 6, 14], [2, 24, 2], [3, 128, 1], [4, 24, 14], [6, 128, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '384.20164', 'commutator_count': 1, 'commutator_label': '192.1541', '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'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1090235, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['384.20164', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 14], [2, 6, 1, 14], [2, 24, 1, 2], [3, 128, 1, 1], [4, 24, 1, 14], [6, 128, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1260, 'exponent': 12, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '768.1090235', 'hash': 1090235, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 1, 3, 2, 2, 2, 2, 2, 2], 'inner_gens': [[6442294336, 127, 79914816, 27909186601, 21196264327, 20717187120, 15335757360, 14856750720, 26951102640], [6442294336, 127, 43591264, 14377673646, 21196264327, 27430179840, 15335757360, 21675270960, 26951102640], [6398703952, 127, 43591264, 21196264327, 27909186601, 15335757360, 20717187120, 26951102640, 14856750720], [38963554583, 127, 27862329911, 14377673646, 21196264327, 27430179840, 15335757360, 21675270960, 26951102640], [6442294336, 127, 14497142105, 14377673646, 21196264327, 27430179840, 15335757360, 21675270960, 26951102640], [44599086016, 127, 20601146464, 14377673646, 21196264327, 27430179840, 15335757360, 21675270960, 26951102640], [6442294336, 127, 27383287984, 14377673646, 21196264327, 27430179840, 15335757360, 21675270960, 26951102640], [33225655696, 127, 14976057904, 14377673646, 21196264327, 27430179840, 15335757360, 21675270960, 26951102640], [6442294336, 127, 21559104304, 14377673646, 21196264327, 27430179840, 15335757360, 21675270960, 26951102640]], 'inner_hash': 20164, 'inner_nilpotent': False, 'inner_order': 384, 'inner_split': True, 'inner_tex': 'C_2^4:S_4', 'inner_used': [1, 3, 4, 6, 8], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 2], [3, 28], [6, 14]], 'label': '768.1090235', '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': True, 'name': 'C2^5: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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 48, 'number_divisions': 48, 'number_normal_subgroups': 37, 'number_subgroup_autclasses': 236, 'number_subgroup_classes': 6845, 'number_subgroups': 41030, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 175], [3, 128], [4, 336], [6, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 84, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[6442294336, 127, 43591264, 14377673646, 21196264327, 27430179840, 15335757360, 126, 7], [6442294463, 127, 43591264, 14377673646, 21196264327, 27430179840, 15335757360, 21675270960, 26951102640], [6442294463, 127, 43591264, 126, 7, 14377673520, 21196264320, 21675270960, 26951102640]], 'outer_group': '336.209', 'outer_hash': 209, 'outer_nilpotent': False, 'outer_order': 336, 'outer_permdeg': 9, 'outer_perms': [21144, 1, 226975], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times \\GL(3,2)', 'pc_rank': None, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [3, 28], [6, 14]], 'representations': {'PC': {'code': '470979877432185466103850158692345232615253586067112816714324791657536', 'gens': [1, 2, 4, 5, 6, 7, 8, 9], 'pres': [9, -2, -2, -3, -2, 2, 2, 2, 2, 2, 181, 46, 218, 1299, 660, 237, 283, 427, 7142, 2939, 7566, 23073, 10041, 17287, 2176, 4561, 45692, 11681, 13634]}, 'Perm': {'d': 14, 'gens': [6442294336, 127, 43591264, 14377673646, 21196264327, 27430179840, 15335757360, 21675270960, 26951102640]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^5:S_4', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
