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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '14520.ba', 'ambient_counter': 27, 'ambient_order': 14520, 'ambient_tex': 'C_5\\times C_{11}^2:D_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 2420, 'characteristic': False, 'core_order': 10, 'counter': 43, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '14520.ba.132.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '132.a1.a1', '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': 132, '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': '110.6', 'subgroup_hash': 6, 'subgroup_order': 110, 'subgroup_tex': 'C_{110}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '14520.ba', 'aut_centralizer_order': None, 'aut_label': '132.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '6.b1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.a1.a1', '66.a1.b1', '66.b1.a1', '66.c1.a1'], 'contains': ['264.a1.a1', '660.a1.a1', '1452.a1.a1'], 'core': '1452.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [4604, -1, 3492, -1, 5640, -1, 6197, -1], 'generators': [212587440, 178774619, 5314686], 'label': '14520.ba.132.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '132.a1.a1', 'projective_image': '1452.22', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '132.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '110.6', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 20, 'aut_gen_orders': [2, 20], 'aut_gens': [[1], [21], [97]], 'aut_group': '40.9', 'aut_hash': 9, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 40, 'aut_permdeg': 11, 'aut_perms': [3719520, 46113], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1], [11, 1, 10, 1], [22, 1, 10, 1], [55, 1, 40, 1], [110, 1, 40, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{20}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '40.9', 'autcent_hash': 9, 'autcent_nilpotent': True, 'autcent_order': 40, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{20}', '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], [5, 1, 4], [10, 1, 4], [11, 1, 10], [22, 1, 10], [55, 1, 40], [110, 1, 40]], 'center_label': '110.6', 'center_order': 110, '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', '5.1', '11.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['2.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1], [11, 1, 10, 1], [22, 1, 10, 1], [55, 1, 40, 1], [110, 1, 40, 1]], 'element_repr_type': 'PC', 'elementary': 110, 'eulerian_function': 1, 'exponent': 110, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[1, 0, 40]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '110.6', 'hash': 6, 'hyperelementary': 110, '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': 40, 'irrQ_dim': 40, 'irrR_degree': 2, 'irrep_stats': [[1, 110]], 'label': '110.6', 'linC_count': 40, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 3, 'linQ_dim': 14, 'linQ_dim_count': 3, 'linR_count': 20, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C110', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 110, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 110, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [5, 4], [10, 4], [11, 10], [22, 10], [55, 40], [110, 40]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [2, 20], 'outer_gen_pows': [0, 0], 'outer_gens': [[21], [97]], 'outer_group': '40.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 40, 'outer_permdeg': 11, 'outer_perms': [3719520, 46113], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{20}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 5, 11], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [10, 2], [40, 2]], 'representations': {'PC': {'code': 3875039, 'gens': [1], 'pres': [3, -2, -5, -11, 6, 34]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [1343, 13320, 2664]}, 'Perm': {'d': 18, 'gens': [355687428096000, 5230697472000, 36288000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [110], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{110}', 'transitive_degree': 110, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [20, 2, 20, 6, 20], 'aut_gens': [[47832241, 131852685, 178774619, 42364499], [96800, 200775886, 22388862, 21915026], [48648144, 45959741, 40913709, 177323829], [115388802, 122827125, 21258843, 99542895], [115539931, 83650977, 2106974, 178774619], [1768294, 15082669, 60400869, 61851659]], 'aut_group': None, 'aut_hash': 7382663879651747906, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 232320, 'aut_permdeg': 136, 'aut_perms': [20014005376341503445578682229803328384308480731858857460427584258207764278303895149409977593219428535941873874142213747329286483535609248604829965439233629791374940184046548018260574170204883557504749514611043170530372083944054545106, 9331112868887371835352271324985092489802687740671295852147066955267031713834658664068636165015734419099355514834136994769562516161322912459658235784800349593431298738963452892098779486708240852289515672042162691036487906179134669905, 26252979859307600031963780229898064099615043130163848628951635633198192578844755205242746319956314138419201520028167814562250347445068678119155187176008229665561452453685526453122232350109698038070523960381067619975875945304802509425, 12028482841753896730961854255257476537183583010172830620202535211583558463269930971542425997893738271412919189316517307145153364328515618908553561988416814726867564686051768110516562936637316175016693850964494112661021419966269467358, 28270386504431988446911878775466862693925536463428392468468849543102713929965618711138637020572917725520306244773473510067254901403604734605560381553542696941477225695891171248834152296762871044951328424160902553831998380059663186732], 'aut_phi_ratio': 66.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 66, 2, 1], [3, 242, 1, 1], [4, 242, 1, 1], [5, 1, 4, 1], [6, 242, 1, 1], [10, 1, 4, 1], [10, 66, 8, 1], [11, 6, 10, 1], [11, 12, 5, 1], [12, 242, 2, 1], [15, 242, 4, 1], [20, 242, 4, 1], [22, 6, 10, 1], [22, 12, 5, 1], [22, 66, 20, 1], [30, 242, 4, 1], [55, 6, 40, 1], [55, 12, 20, 1], [60, 242, 8, 1], [110, 6, 40, 1], [110, 12, 20, 1], [110, 66, 80, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{11}^2.C_6.C_{10}.C_4.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 660, 'autcentquo_group': None, 'autcentquo_hash': 8486544706216729924, 'autcentquo_nilpotent': False, 'autcentquo_order': 14520, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{11}^2:(C_5\\times D_{12})', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 66, 2], [3, 242, 1], [4, 242, 1], [5, 1, 4], [6, 242, 1], [10, 1, 4], [10, 66, 8], [11, 6, 10], [11, 12, 5], [12, 242, 2], [15, 242, 4], [20, 242, 4], [22, 6, 10], [22, 12, 5], [22, 66, 20], [30, 242, 4], [55, 6, 40], [55, 12, 20], [60, 242, 8], [110, 6, 40], [110, 12, 20], [110, 66, 80]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '1452.22', 'commutator_count': 1, 'commutator_label': '726.7', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1', '11.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 27, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2904.p', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 66, 1, 2], [3, 242, 1, 1], [4, 242, 1, 1], [5, 1, 4, 1], [6, 242, 1, 1], [10, 1, 4, 1], [10, 66, 4, 2], [11, 6, 5, 2], [11, 12, 5, 1], [12, 242, 2, 1], [15, 242, 4, 1], [20, 242, 4, 1], [22, 6, 5, 2], [22, 12, 5, 1], [22, 66, 10, 2], [30, 242, 4, 1], [55, 6, 20, 2], [55, 12, 20, 1], [60, 242, 8, 1], [110, 6, 20, 2], [110, 12, 20, 1], [110, 66, 40, 2]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 216, 'exponent': 660, 'exponents_of_order': [3, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[6, 0, 80], [12, 0, 20]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '7260.bh', 'hash': 6269800531405236361, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [2, 6, 11, 11], 'inner_gens': [[47832241, 132124693, 177323829, 40913709], [131576778, 131852685, 177323829, 3222352], [48001278, 54866358, 178774619, 42364499], [48001278, 54713293, 178774619, 42364499]], 'inner_hash': 22, 'inner_nilpotent': False, 'inner_order': 1452, 'inner_split': True, 'inner_tex': 'C_{11}^2:D_6', 'inner_used': [1, 2, 3], 'irrC_degree': 6, 'irrQ_degree': 240, 'irrQ_dim': 240, 'irrR_degree': 12, 'irrep_stats': [[1, 20], [2, 25], [6, 200], [12, 50]], 'label': '14520.ba', 'linC_count': 80, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 36, 'linQ_degree_count': 16, 'linQ_dim': 36, 'linQ_dim_count': 16, 'linR_count': 600, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C5*C11^2:D12', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 295, 'number_divisions': 32, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 70, 'number_subgroup_classes': 102, 'number_subgroups': 4732, 'old_label': None, 'order': 14520, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 133], [3, 242], [4, 242], [5, 4], [6, 242], [10, 532], [11, 120], [12, 484], [15, 968], [20, 968], [22, 1440], [30, 968], [55, 480], [60, 1936], [110, 5760]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [2, 4, 20], 'outer_gen_pows': [1499553, 1771562, 106969989], 'outer_gens': [[1356047, 131852685, 139158757, 138991051], [5314801, 132124693, 177323829, 40913709], [105623019, 84292519, 118694643, 159287459]], 'outer_group': '160.179', 'outer_hash': 179, 'outer_nilpotent': True, 'outer_order': 160, 'outer_permdeg': 13, 'outer_perms': [479042640, 91440, 1041506730], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_{20}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 42, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 5], [8, 3], [16, 1], [30, 4], [60, 4], [120, 4], [240, 4]], 'representations': {'PC': {'code': '203681111293444972460283128482036178131865571636410458920950209410300199', 'gens': [1, 3, 6, 7], 'pres': [7, -2, -5, -2, -2, -3, -11, 11, 14, 2312, 58, 2803, 80, 2804, 554405, 55459, 30266, 1419, 58806, 7076, 321, 17674]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 31893194831329726, 1154774342334730, 16267516240797404, 5063783017457019, 33418192856010432, 8340314952409832]}, 'GLZq': {'d': 2, 'q': 121, 'gens': [106969989, 47832174, 138991051, 286649, 212587440, 22709633, 1771681]}, 'Perm': {'d': 42, 'gens': [134145892997467801986078850885665177138706254643200, 90, 1008277542644983874896799129187210237147414059942400, 20400952242353932147493255765951303329290612480, 90720, 115048533495526564162974779163712794019491481747200, 2468669465003927770184557017568109535440169103440]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times C_{11}^2:D_{12}', 'transitive_degree': 660, 'wreath_data': None, 'wreath_product': False}
