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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '21600.bg', 'ambient_counter': 33, 'ambient_order': 21600, 'ambient_tex': 'C_5\\times A_4\\times A_6', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': False, 'core_order': 5, 'counter': 83, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '21600.bg.180.a1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '180.a1.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': 180, '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': '120.43', 'subgroup_hash': 43, 'subgroup_order': 120, 'subgroup_tex': 'C_{10}\\times A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '21600.bg', 'aut_centralizer_order': None, 'aut_label': '180.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1080.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['36.a1.b1', '90.a1.b1', '90.b1.b1', '90.e1.b1'], 'contains': ['360.j1.b1', '540.d1.a2', '720.a1.b1', '900.a1.b1'], 'core': '4320.a1.a1', 'coset_action_label': None, 'count': 45, 'diagramx': [4901, -1, 2368, -1, 5840, -1, 1020, -1], 'generators': [274467916800, 4072532, 8431920, 274476349134, 274467917191], 'label': '21600.bg.180.a1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '45.b1.b1', 'old_label': '180.a1.b1', 'projective_image': '4320.bh', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '180.a1.b1', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '30.4', '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, 4, 4, 2, 6], 'aut_gens': [[1, 6, 12], [1, 6, 108], [67, 6, 108], [7, 6, 84], [1, 6, 36], [71, 66, 108], [61, 60, 114]], 'aut_group': '96.186', 'aut_hash': 186, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [16, 11536, 5169, 18, 5040, 15256], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 4, 2, 1], [5, 1, 4, 1], [6, 4, 2, 1], [10, 1, 4, 1], [10, 3, 4, 2], [15, 4, 8, 1], [30, 4, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 4, 2], [5, 1, 4], [6, 4, 2], [10, 1, 4], [10, 3, 8], [15, 4, 8], [30, 4, 8]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 43, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['2.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 4, 2, 1], [5, 1, 4, 1], [6, 4, 2, 1], [10, 1, 4, 1], [10, 3, 4, 2], [15, 4, 8, 1], [30, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 30, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '120.43', 'hash': 43, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[1, 60, 18], [67, 6, 12], [7, 6, 12]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 6, 'irrep_stats': [[1, 30], [3, 10]], 'label': '120.43', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 3, 'linQ_dim': 7, 'linQ_dim_count': 3, 'linR_count': 18, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C10*A4', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 40, 'number_divisions': 12, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 24, 'number_subgroups': 52, 'old_label': None, 'order': 120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 8], [5, 4], [6, 8], [10, 28], [15, 32], [30, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[5, 60, 54], [1, 6, 84]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [3, 2], [4, 2], [8, 2], [12, 2]], 'representations': {'PC': {'code': 7121650444183381059, 'gens': [1, 3, 4], 'pres': [5, -2, -3, -2, 2, -5, 10, 902, 502, 363, 788, 58]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [656967451, 1230157177, 857494092, 1714988184, 1907820797]}, 'Perm': {'d': 11, 'gens': [1, 403200, 870, 3669120, 7983360]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [30], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}\\times A_4', 'transitive_degree': 30, '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': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [30, 6, 24, 6, 6], 'aut_gens': [[87657292800, 274467916800, 100111334400, 12454041936, 12454041626, 100122225840], [186810624000, 87657292800, 267761894400, 262013875805, 262013875614, 267765966720], [274467916800, 186810624000, 267761894400, 12454041856, 12454042247, 267772785840], [274467916800, 186810624000, 6706022400, 181062605284, 181062604826, 6714454320], [274467916800, 87657292800, 12454041600, 100111334862, 100111334534, 12464933040], [186810624000, 87657292800, 267761894400, 262013875891, 262013875913, 267772785840]], 'aut_group': None, 'aut_hash': 3276841175708577286, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 138240, 'aut_permdeg': 57, 'aut_perms': [28864642645510784913822712568338074894966545558538332801404185879885243627209, 406754175148224496264176332651967572413958800451502465252211691197479430041, 24970002017494506648920163229180824837034396418307601982675276446746129142256, 7083116372180273981603048098034343163928645393911102376583004552491652171634, 36569130611034786872933803060699834655504437890301521749663823280208361801470], 'aut_phi_ratio': 24.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 45, 1, 1], [2, 135, 1, 1], [3, 4, 2, 1], [3, 40, 2, 1], [3, 160, 4, 1], [4, 90, 1, 1], [4, 270, 1, 1], [5, 1, 4, 1], [5, 72, 2, 1], [5, 72, 8, 1], [6, 120, 2, 1], [6, 180, 2, 1], [10, 3, 4, 1], [10, 45, 4, 1], [10, 135, 4, 1], [10, 216, 2, 1], [10, 216, 8, 1], [12, 360, 2, 1], [15, 4, 8, 1], [15, 40, 8, 1], [15, 160, 16, 1], [15, 288, 4, 1], [15, 288, 16, 1], [20, 90, 4, 1], [20, 270, 4, 1], [30, 120, 8, 1], [30, 180, 8, 1], [60, 360, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times A_6.C_2^2\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '34560.m', 'autcentquo_hash': 7190829667096474948, 'autcentquo_nilpotent': False, 'autcentquo_order': 34560, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times S_6:C_2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 45, 1], [2, 135, 1], [3, 4, 2], [3, 40, 2], [3, 160, 4], [4, 90, 1], [4, 270, 1], [5, 1, 4], [5, 72, 10], [6, 120, 2], [6, 180, 2], [10, 3, 4], [10, 45, 4], [10, 135, 4], [10, 216, 10], [12, 360, 2], [15, 4, 8], [15, 40, 8], [15, 160, 16], [15, 288, 20], [20, 90, 4], [20, 270, 4], [30, 120, 8], [30, 180, 8], [60, 360, 8]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '4320.bh', 'commutator_count': 1, 'commutator_label': '1440.5845', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1', '360.118'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 33, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['360.118', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 45, 1, 1], [2, 135, 1, 1], [3, 4, 2, 1], [3, 40, 1, 2], [3, 160, 2, 2], [4, 90, 1, 1], [4, 270, 1, 1], [5, 1, 4, 1], [5, 72, 2, 1], [5, 72, 4, 2], [6, 120, 1, 2], [6, 180, 2, 1], [10, 3, 4, 1], [10, 45, 4, 1], [10, 135, 4, 1], [10, 216, 2, 1], [10, 216, 4, 2], [12, 360, 2, 1], [15, 4, 8, 1], [15, 40, 4, 2], [15, 160, 8, 2], [15, 288, 4, 1], [15, 288, 8, 2], [20, 90, 4, 1], [20, 270, 4, 1], [30, 120, 4, 2], [30, 180, 8, 1], [60, 360, 8, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1272, 'exponent': 60, 'exponents_of_order': [5, 3, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[15, 0, 8], [24, 0, 8], [27, 0, 4], [30, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '21600.bg', 'hash': 1049615865346914016, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 2, 3, 12, 6, 3], 'inner_gens': [[87657292800, 274467916800, 174356582400, 181062605136, 181062604826, 174367473840], [87657292800, 274467916800, 267761894400, 93405312336, 93405312026, 267772785840], [186810624000, 87657292800, 100111334400, 181062605136, 181062604826, 100122225840], [274467916800, 186810624000, 267761894400, 12454041936, 12454042287, 267772785840], [274467916800, 186810624000, 267761894400, 12454041742, 12454041626, 267772785840], [186810624000, 87657292800, 100111334400, 181062605136, 181062604826, 100122225840]], 'inner_hash': 8888903546687000962, 'inner_nilpotent': False, 'inner_order': 4320, 'inner_split': False, 'inner_tex': 'A_4\\times A_6', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 15, 'irrQ_degree': 60, 'irrQ_dim': 60, 'irrR_degree': 30, 'irrep_stats': [[1, 15], [3, 5], [5, 30], [8, 30], [9, 15], [10, 15], [15, 10], [24, 10], [27, 5], [30, 5]], 'label': '21600.bg', 'linC_count': 144, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 2, 'linQ_dim': 12, 'linQ_dim_count': 2, 'linR_count': 12, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C5*A4*A6', '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, 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': 30, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 140, 'number_divisions': 39, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 223, 'number_subgroup_classes': 322, 'number_subgroups': 22252, 'old_label': None, 'order': 21600, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 183], [3, 728], [4, 360], [5, 724], [6, 600], [10, 2892], [12, 720], [15, 8672], [20, 1440], [30, 2400], [60, 2880]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 352, 425, 0], 'outer_gens': [[186810624000, 274467916800, 262013875200, 174356582736, 174356582426, 262024766640], [87657292800, 274467916800, 100111334400, 12454041908, 12454041607, 100122225840], [87657292800, 274467916800, 100111334400, 12454042196, 12454042287, 100122225840], [87657292800, 274467916800, 100111334400, 12454041936, 12454041626, 100119766320]], 'outer_group': '32.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 5040, 120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 1], [5, 2], [8, 1], [9, 1], [10, 3], [12, 1], [15, 2], [16, 1], [18, 1], [20, 3], [27, 1], [30, 1], [32, 3], [36, 1], [40, 3], [48, 1], [60, 2], [64, 2], [72, 1], [80, 1], [96, 2], [108, 1], [120, 1]], 'representations': {'Perm': {'d': 15, 'gens': [87657292800, 274467916800, 100111334400, 12454041936, 12454041626, 100122225840]}}, 'schur_multiplier': [2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times A_4\\times A_6', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
