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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '960.5544', 'ambient_counter': 5544, 'ambient_order': 960, 'ambient_tex': 'C_4^2\\times C_{60}', 'central': True, 'central_factor': False, 'centralizer_order': 960, 'characteristic': False, 'core_order': 2, 'counter': 39, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '960.5544.480.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '480.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '480.919', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 919, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 480, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_4\\times C_{60}', 'simple': True, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.5544', 'aut_centralizer_order': 98304, 'aut_label': '480.a1', 'aut_quo_index': 1, 'aut_stab_index': 7, 'aut_weyl_group': '1.1', 'aut_weyl_index': 688128, 'centralizer': '1.a1', 'complements': [], 'conjugacy_class_count': 7, 'contained_in': ['96.a1', '160.a1', '240.a1', '240.b1'], 'contains': ['960.a1'], 'core': '480.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [2293, 2293, 3986, 3986], 'generators': [8], 'label': '960.5544.480.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '480.a1', 'normal_contained_in': ['96.a1', '160.a1', '240.a1', '240.b1'], 'normal_contains': ['960.a1'], 'normalizer': '1.a1', 'old_label': '480.a1', 'projective_image': '480.919', 'quotient_action_image': '1.1', 'quotient_action_kernel': '480.919', 'quotient_action_kernel_order': 480, 'quotient_fusion': None, 'short_label': '480.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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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}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '960.5544', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 168, 'aut_gen_orders': [8, 4, 12, 2], 'aut_gens': [[1, 4, 16], [246, 487, 940], [724, 13, 902], [250, 12, 761], [1, 249, 431]], 'aut_group': None, 'aut_hash': 4847960479678897568, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 688128, 'aut_permdeg': 62, 'aut_perms': [21859475432173915706105217376216563584224874433939889767599492369552748701570653440350, 7114618011135651659067332664736226122269463378531714656567139394466911939070025212538, 27934282479768071593676230411194567525022905776140052237742082926914391945967944898670, 108201924417147595937121572752716500107817647784175678429589475743173987900451025], 'aut_phi_ratio': 2688.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [3, 1, 2, 1], [4, 1, 56, 1], [5, 1, 4, 1], [6, 1, 14, 1], [10, 1, 28, 1], [12, 1, 112, 1], [15, 1, 8, 1], [20, 1, 224, 1], [30, 1, 56, 1], [60, 1, 448, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_4\\times C_2^9.\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': None, 'autcent_hash': 4847960479678897568, 'autcent_nilpotent': False, 'autcent_order': 688128, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times C_4\\times C_2^9.\\PSL(2,7)', '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, 7], [3, 1, 2], [4, 1, 56], [5, 1, 4], [6, 1, 14], [10, 1, 28], [12, 1, 112], [15, 1, 8], [20, 1, 224], [30, 1, 56], [60, 1, 448]], 'center_label': '960.5544', 'center_order': 960, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 5544, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 3], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 1], [4, 1, 2, 28], [5, 1, 4, 1], [6, 1, 2, 7], [10, 1, 4, 7], [12, 1, 4, 28], [15, 1, 8, 1], [20, 1, 8, 28], [30, 1, 8, 7], [60, 1, 16, 28]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 403, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '120.47', 'hash': 5544, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 4, 16], [1, 4, 16], [1, 4, 16]], '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': None, 'irrep_stats': [[1, 960]], 'label': '960.5544', 'linC_count': None, 'linC_degree': 3, '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2*C60', 'ngens': 8, 'nilpotency_class': 1, 'nilpotent': True, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 960, 'number_divisions': 144, 'number_normal_subgroups': 516, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 516, 'number_subgroups': 516, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 56], [5, 4], [6, 14], [10, 28], [12, 112], [15, 8], [20, 224], [30, 56], [60, 448]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [14, 6, 8, 6], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[734, 483, 467], [494, 485, 472], [250, 251, 374], [6, 15, 16]], 'outer_group': None, 'outer_hash': 4847960479678897568, 'outer_nilpotent': False, 'outer_order': 688128, 'outer_permdeg': 62, 'outer_perms': [18323467192080420847865184746780267647058081712378625138717287388011060860438893189633, 30505382893737326537288456274618206504228885190498453908300141709696960641670456213308, 27950459161574032373702127174558410875782581435025634410955094802175328307973783238898, 5084313602432589715032155592123388736245639873903716801282301939387412048699993518399], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times C_4\\times C_2^9.\\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 4, 3, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 36], [4, 36], [8, 36], [16, 28]], 'representations': {'PC': {'code': 1922550926486735077850284039, 'gens': [1, 3, 5], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -5, 16, 66, 116, 141, 222]}, 'GLZN': {'d': 2, 'p': 55, 'gens': [273106, 5324032, 3826648, 1089291, 1996534, 2662016, 166409, 5656784]}, 'Perm': {'d': 20, 'gens': [377740048637952000, 4097379686400, 127008000, 10080, 96, 122000787836928000, 1313901388800, 40279680]}}, 'schur_multiplier': [4, 4, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4, 60], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2\\times C_{60}', 'transitive_degree': 960, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '480.919', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [4, 4, 4, 6], 'aut_gens': [[1, 2, 8], [245, 247, 13], [1, 242, 475], [1, 7, 111], [1, 121, 90]], 'aut_group': None, 'aut_hash': 8596656399186838937, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 30, 'aut_perms': [159163210500281568292638327290412, 155759454552747412435426033126130, 8848246704567947517120964212813, 63984318851403707379571767564554], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 1, 4, 1], [3, 1, 2, 1], [4, 1, 24, 1], [5, 1, 4, 1], [6, 1, 6, 1], [6, 1, 8, 1], [10, 1, 12, 1], [10, 1, 16, 1], [12, 1, 48, 1], [15, 1, 8, 1], [20, 1, 96, 1], [30, 1, 24, 1], [30, 1, 32, 1], [60, 1, 192, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_4\\times C_2^6.S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': None, 'autcent_hash': 8596656399186838937, 'autcent_nilpotent': False, 'autcent_order': 12288, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times C_4\\times C_2^6.S_4', '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, 7], [3, 1, 2], [4, 1, 24], [5, 1, 4], [6, 1, 14], [10, 1, 28], [12, 1, 48], [15, 1, 8], [20, 1, 96], [30, 1, 56], [60, 1, 192]], 'center_label': '480.919', 'center_order': 480, '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', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 919, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['4.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 1], [4, 1, 2, 12], [5, 1, 4, 1], [6, 1, 2, 7], [10, 1, 4, 7], [12, 1, 4, 12], [15, 1, 8, 1], [20, 1, 8, 12], [30, 1, 8, 7], [60, 1, 16, 12]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 2821, 'exponent': 60, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '120.47', 'hash': 919, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 8], [1, 2, 8], [1, 2, 8]], '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, 480]], 'label': '480.919', 'linC_count': None, 'linC_degree': 3, '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*C4*C60', 'ngens': 7, 'nilpotency_class': 1, 'nilpotent': True, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 480, 'number_divisions': 80, 'number_normal_subgroups': 216, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 216, 'number_subgroups': 216, 'old_label': None, 'order': 480, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 24], [5, 4], [6, 14], [10, 28], [12, 48], [15, 8], [20, 96], [30, 56], [60, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [4, 4, 3, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[5, 124, 134], [1, 363, 380], [241, 360, 254], [245, 120, 227]], 'outer_group': None, 'outer_hash': 8596656399186838937, 'outer_nilpotent': False, 'outer_order': 12288, 'outer_permdeg': 30, 'outer_perms': [118014485420116202904712819931342, 219182020528196612718537792691205, 72890964295058794160865053486462, 73811293597240036471345672796639], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times C_4\\times C_2^6.S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 4, 3, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 20], [8, 20], [16, 12]], 'representations': {'PC': {'code': 245743619577238945799, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -2, -2, -2, -3, -5, 36, 80, 102, 166]}, 'GLZN': {'d': 2, 'p': 40, 'gens': [1088017, 2513229, 576009, 2312611, 1344021, 192003, 64321]}, 'Perm': {'d': 18, 'gens': [4097379686400, 127008000, 355687428096000, 10080, 96, 1313901388800, 40279680]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 60], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4\\times C_{60}', 'transitive_degree': 480, 'wreath_data': None, 'wreath_product': False}
