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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '960.1492', 'ambient_counter': 1492, 'ambient_order': 960, 'ambient_tex': 'C_{20}.(S_3\\times Q_8)', 'central': False, 'central_factor': False, 'centralizer_order': 160, 'characteristic': False, 'core_order': 4, 'counter': 160, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.1492.120.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '120.b1.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': 120, '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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.1492', 'aut_centralizer_order': 2560, 'aut_label': '120.b1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '4.2', 'aut_weyl_index': 7680, 'centralizer': '6.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['24.b1.a1', '40.b1.a1', '60.b1.a1', '60.h1.a1', '60.i1.a1'], 'contains': ['240.a1.a1', '240.d1.a1', '240.d1.b1'], 'core': '240.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [3095, -1, 2952, -1, 3168, -1, 2811, -1], 'generators': [1, 96], 'label': '960.1492.120.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '40.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '120.b1.a1', 'projective_image': '240.150', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '120.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_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, 3], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', 'transitive_degree': 8, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [4, 4, 10, 2, 4, 2, 4, 2, 4, 60], 'aut_gens': [[1, 2, 8, 192], [1, 102, 904, 384], [1, 54, 664, 384], [1, 146, 764, 192], [101, 2, 732, 768], [5, 98, 200, 384], [101, 2, 588, 768], [97, 98, 572, 384], [133, 98, 88, 192], [5, 150, 200, 768], [133, 54, 680, 192]], 'aut_group': None, 'aut_hash': 7312598050044633167, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 30720, 'aut_permdeg': 60, 'aut_perms': [4085376209459534433156274019623716740216037878124371908288454062575034354662007099, 5331172998145928479304943319224266061589868257670654782586834077597980616907525179, 4374493838045490281984765191133184710232789603077744729015061678818585827657686352, 1130178649371894214460436250454993028701798548296485849123962213991943927716258519, 3963471258895845192099554690066901661861206531578116379809066102532961011423817608, 8320662329573915686163166383730163733843141925931994237238113893847521272012616919, 2253706349260025362984679028918535224493481674690163651787497539620971876628517073, 5642307544213165901429613800186801533055572705503944039010526210042248997323527040, 6182411229727728256358957738110781595272087710794015890986119348220560729913731831, 5785687515210142002271313173001011991454529509418839241719785287203200611119167416], 'aut_phi_ratio': 120.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 2, 1, 2], [4, 6, 2, 1], [4, 8, 1, 1], [4, 12, 1, 1], [4, 20, 2, 1], [4, 24, 1, 1], [4, 60, 2, 1], [5, 2, 2, 1], [6, 2, 1, 3], [8, 20, 2, 1], [8, 60, 2, 1], [10, 2, 2, 3], [12, 4, 1, 2], [12, 8, 2, 1], [12, 40, 2, 1], [15, 4, 2, 1], [20, 4, 2, 2], [20, 8, 4, 1], [20, 12, 4, 2], [20, 24, 4, 1], [24, 20, 4, 1], [30, 4, 2, 3], [60, 8, 2, 2], [60, 8, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{15}:(C_2^2.C_2^6.C_2^3)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '480.1197', 'autcentquo_hash': 1197, 'autcentquo_nilpotent': False, 'autcentquo_order': 480, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 1], [4, 2, 2], [4, 6, 2], [4, 8, 1], [4, 12, 1], [4, 20, 2], [4, 24, 1], [4, 60, 2], [5, 2, 2], [6, 2, 3], [8, 20, 2], [8, 60, 2], [10, 2, 6], [12, 4, 2], [12, 8, 2], [12, 40, 2], [15, 4, 2], [20, 4, 4], [20, 8, 4], [20, 12, 8], [20, 24, 4], [24, 20, 4], [30, 4, 6], [60, 8, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '240.150', 'commutator_count': 2, 'commutator_label': '120.31', 'complements_known': True, 'complete': False, 'complex_characters_known': True, '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': 1492, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 2, 1, 2], [4, 6, 1, 2], [4, 8, 1, 1], [4, 12, 1, 1], [4, 20, 2, 1], [4, 24, 1, 1], [4, 60, 1, 2], [5, 2, 2, 1], [6, 2, 1, 3], [8, 20, 2, 1], [8, 60, 2, 1], [10, 2, 2, 3], [12, 4, 1, 2], [12, 8, 2, 1], [12, 40, 2, 1], [15, 4, 2, 1], [20, 4, 2, 2], [20, 8, 4, 1], [20, 12, 2, 2], [20, 12, 4, 1], [20, 24, 4, 1], [24, 20, 4, 1], [30, 4, 2, 3], [60, 8, 2, 2], [60, 8, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 8064, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '120.42', 'hash': 1492, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 2, 12, 5], 'inner_gens': [[1, 6, 140, 192], [5, 2, 152, 192], [69, 50, 8, 768], [1, 2, 392, 192]], 'inner_hash': 150, 'inner_nilpotent': False, 'inner_order': 240, 'inner_split': True, 'inner_tex': 'D_6:D_{10}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 34], [4, 35], [8, 4]], 'label': '960.1492', 'linC_count': 320, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 126, 'linQ_dim': 14, 'linQ_dim_count': 104, 'linR_count': 8, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C20.(S3*Q8)', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 40, 'number_characteristic_subgroups': 60, 'number_conjugacy_classes': 81, 'number_divisions': 43, 'number_normal_subgroups': 66, 'number_subgroup_autclasses': 176, 'number_subgroup_classes': 192, 'number_subgroups': 984, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 220], [5, 4], [6, 6], [8, 160], [10, 12], [12, 104], [15, 8], [20, 240], [24, 80], [30, 24], [60, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 136, 192], [97, 2, 8, 192], [1, 2, 104, 192], [5, 2, 8, 192], [1, 6, 8, 192], [1, 2, 8, 384]], 'outer_group': '128.2319', 'outer_hash': 2319, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 14, 'outer_perms': [362880, 6227020800, 39916800, 127, 5040, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 10], [8, 11], [16, 6]], 'representations': {'PC': {'code': 4167902786921102508603304010772777000506800431890439, 'gens': [1, 2, 4, 8], 'pres': [8, -2, -2, -2, 2, -2, -2, -3, -5, 32, 97, 41, 4483, 2443, 91, 3204, 2252, 116, 7685, 141, 7174, 6175]}, 'Perm': {'d': 24, 'gens': [27083077961278184254824, 55085780431078738750081, 4557190755194842423680, 83436746068083220606080, 4302991486080, 55085780429570322432000, 3, 6504]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}.(S_3\\times Q_8)', 'transitive_degree': 960, 'wreath_data': None, 'wreath_product': False}
