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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '39930.c', 'ambient_counter': 3, 'ambient_order': 39930, 'ambient_tex': 'C_5\\times C_{11}\\wr S_3', 'central': False, 'central_factor': False, 'centralizer_order': 1210, 'characteristic': False, 'core_order': 55, 'counter': 37, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '39930.c.363.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '363.a1', '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': 363, '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': '39930.c', 'aut_centralizer_order': None, 'aut_label': '363.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '33.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['33.a1', '33.b1', '121.a1'], 'contains': ['726.a1', '1815.a1', '3993.a1'], 'core': '726.a1', 'coset_action_label': None, 'count': 33, 'diagramx': [7279, -1, 7220, -1], 'generators': [1771681, 157669018, 143496522], 'label': '39930.c.363.a1', 'mobius_quo': None, 'mobius_sub': 11, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '33.a1', 'old_label': '363.a1', 'projective_image': '726.5', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '363.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': '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': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '110.6', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [12, 10, 110, 22, 20], 'aut_gens': [[157669049, 107403653, 177644600, 162983682], [204841060, 127364649, 41234480, 173459955], [106434972, 9701895, 40746003, 122573179], [68972349, 68804917, 177644600, 25931986], [146379514, 107458466, 79720323, 28024231], [185186238, 87008993, 177156123, 67975517]], 'aut_group': None, 'aut_hash': 3408281614888932655, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 290400, 'aut_permdeg': 47, 'aut_perms': [185900222615568644856876809332618281893810635637220250490113, 76563585737373329395084132466750670189248870604569237947395, 38269227270162736545075254955030700535938036681952052511386, 185898347622074758438859139742570083473162740477562973640765, 75873590595994049670962049582514064516819481628702510255739], 'aut_phi_ratio': 30.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 33, 1, 1], [3, 242, 1, 1], [5, 1, 4, 1], [10, 33, 4, 1], [11, 1, 10, 1], [11, 3, 10, 1], [11, 3, 100, 1], [11, 6, 5, 1], [11, 6, 10, 1], [11, 6, 50, 1], [11, 6, 100, 1], [15, 242, 4, 1], [22, 33, 10, 2], [22, 33, 100, 1], [33, 242, 10, 1], [55, 1, 40, 1], [55, 3, 40, 1], [55, 3, 400, 1], [55, 6, 20, 1], [55, 6, 40, 1], [55, 6, 200, 1], [55, 6, 400, 1], [110, 33, 40, 2], [110, 33, 400, 1], [165, 242, 40, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{11}^2.C_{15}.C_{10}.C_2^4', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '7260.bg', 'autcentquo_hash': 7237787727408842954, 'autcentquo_nilpotent': False, 'autcentquo_order': 7260, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{11}^2:(S_3\\times C_{10})', 'cc_stats': [[1, 1, 1], [2, 33, 1], [3, 242, 1], [5, 1, 4], [10, 33, 4], [11, 1, 10], [11, 3, 110], [11, 6, 165], [15, 242, 4], [22, 33, 120], [33, 242, 10], [55, 1, 40], [55, 3, 440], [55, 6, 660], [110, 33, 480], [165, 242, 40]], 'center_label': '55.2', 'center_order': 55, 'central_product': True, 'central_quotient': '726.5', 'commutator_count': 1, 'commutator_label': '363.2', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '3.1', '5.1', '11.1', '11.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['5.1', 1], ['726.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 33, 1, 1], [3, 242, 1, 1], [5, 1, 4, 1], [10, 33, 4, 1], [11, 1, 10, 1], [11, 3, 10, 11], [11, 6, 5, 1], [11, 6, 10, 16], [15, 242, 4, 1], [22, 33, 10, 12], [33, 242, 10, 1], [55, 1, 40, 1], [55, 3, 40, 11], [55, 6, 20, 1], [55, 6, 40, 16], [110, 33, 40, 12], [165, 242, 40, 1]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 2592, 'exponent': 330, 'exponents_of_order': [3, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[3, 0, 800], [6, 0, 600]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '39930.c', 'hash': 164722554699628861, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [22, 3, 11, 11], 'inner_gens': [[157669049, 107946943, 178453848, 162983682], [145464875, 107403653, 79720323, 26908853], [158639348, 10441689, 177644600, 162983682], [157669049, 107556718, 177644600, 162983682]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 726, 'inner_split': True, 'inner_tex': 'C_{11}^2:S_3', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 120, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 110], [2, 55], [3, 1100], [6, 825]], 'label': '39930.c', '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': 'C5*C11wrS3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 28, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 2090, 'number_divisions': 90, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 68, 'number_subgroup_classes': 192, 'number_subgroups': 2520, 'old_label': None, 'order': 39930, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 33], [3, 242], [5, 4], [10, 132], [11, 1330], [15, 968], [22, 3960], [33, 2420], [55, 5320], [110, 15840], [165, 9680]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 20, 'outer_gen_orders': [2, 10, 20], 'outer_gen_pows': [107403653, 107403653, 1771562], 'outer_gens': [[205318889, 107403653, 41234480, 46396013], [48991432, 107403653, 118694643, 182959319], [79720354, 107403653, 42043728, 184242415]], 'outer_group': '400.201', 'outer_hash': 201, 'outer_nilpotent': True, 'outer_order': 400, 'outer_permdeg': 18, 'outer_perms': [1307674368000, 355693696248960, 13451961633], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{10}\\times C_{20}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 38, 'pgroup': 0, 'primary_abelian_invariants': [2, 5, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1], [10, 2], [20, 1], [30, 23], [40, 2], [60, 16], [80, 1], [120, 23], [240, 16]], 'representations': {'PC': {'code': '94865677111217709399317494488190046815441091939', 'gens': [1, 3, 4, 5], 'pres': [6, 2, 11, 3, 11, 5, 11, 12, 78014, 132074, 887043, 3975, 20986, 178, 60497]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 36596619999581411, 28745646345613891, 42771567939938742, 19063078487671420, 17298483428108842]}, 'GLZq': {'d': 2, 'q': 121, 'gens': [106969989, 47832174, 1771617, 41081349, 196643382, 1771681]}, 'Perm': {'d': 38, 'gens': [29462864638184677479536993474525075498035920, 90, 96402800143457235296762783534192263967969760, 213648602383098892524023385440526594412018560, 57807522871758521146359618321510186331893840, 72636314234587979715969788280020836147200]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [110], '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}\\wr S_3', 'transitive_degree': 165, 'wreath_data': None, 'wreath_product': False}
