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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1008.598', 'ambient_counter': 598, 'ambient_order': 1008, 'ambient_tex': 'D_{42}:D_6', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 28, 'counter': 45, 'cyclic': False, 'direct': None, 'hall': 14, 'label': '1008.598.9.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '9.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': 9, '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': '112.31', 'subgroup_hash': 31, 'subgroup_order': 112, 'subgroup_tex': 'D_4\\times D_7', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1008.598', 'aut_centralizer_order': 4, 'aut_label': '9.a1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '336.216', 'aut_weyl_index': 36, 'centralizer': '504.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1.a1', '3.b1.a1'], 'contains': ['18.a1.a1', '18.b1.a1', '18.c1.a1', '18.d1.a1', '18.e1.a1', '18.f1.a1', '18.g1.a1', '63.a1.a1'], 'core': '36.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [6814, -1, 7797, -1, 6483, -1, 5428, -1], 'generators': [1, 84, 168, 2, 48], 'label': '1008.598.9.a1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '9.a1.a1', 'projective_image': '504.172', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '9.a1.a1', 'subgroup_fusion': None, 'weyl_group': '56.12'}
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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': 84, 'aut_gen_orders': [2, 2, 6, 28], 'aut_gens': [[1, 2, 4], [57, 58, 60], [57, 2, 4], [1, 2, 68], [97, 30, 4]], 'aut_group': '672.1093', 'aut_hash': 1093, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 672, 'aut_permdeg': 13, 'aut_perms': [120, 1, 44357040, 2402270934], 'aut_phi_ratio': 14.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [2, 7, 2, 1], [2, 14, 2, 1], [4, 2, 1, 1], [4, 14, 1, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 4, 6, 1], [28, 4, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '84.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 84, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [2, 7, 2], [2, 14, 2], [4, 2, 1], [4, 14, 1], [7, 2, 3], [14, 2, 3], [14, 4, 6], [28, 4, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '56.12', 'commutator_count': 1, 'commutator_label': '14.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [2, 7, 1, 2], [2, 14, 1, 2], [4, 2, 1, 1], [4, 14, 1, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 4, 3, 2], [28, 4, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 672, 'exponent': 28, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 1, 3]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '56.12', 'hash': 31, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 2, 14], 'inner_gens': [[1, 2, 52], [1, 2, 60], [65, 58, 4]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': True, 'inner_tex': 'C_2\\times D_{14}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 14], [4, 3]], 'label': '112.31', 'linC_count': 27, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 8, 'linR_count': 27, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4*D7', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 25, 'number_divisions': 15, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 34, 'number_subgroup_classes': 54, 'number_subgroups': 220, 'old_label': None, 'order': 112, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 47], [4, 16], [7, 6], [14, 30], [28, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [28, 0], 'outer_gens': [[1, 30, 4], [57, 2, 20]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [6, 4], [12, 1]], 'representations': {'PC': {'code': 22553896189964738693, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -7, 782, 457, 42, 2083, 58, 2404]}, 'GLZN': {'d': 2, 'p': 21, 'gens': [142311, 9325, 9269, 74096, 59004]}, 'Perm': {'d': 11, 'gens': [368047, 1, 16, 7, 4364760]}}, 'schur_multiplier': [2, 2, 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': 'D_4\\times D_7', 'transitive_degree': 28, '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': 42, 'aut_gen_orders': [2, 6, 14, 6, 6, 6, 6], 'aut_gens': [[1, 2, 4, 336], [113, 98, 668, 672], [281, 218, 220, 672], [225, 98, 956, 672], [57, 314, 44, 336], [281, 26, 428, 672], [113, 74, 932, 672], [1, 122, 428, 672]], 'aut_group': None, 'aut_hash': 6486938177741458836, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12096, 'aut_permdeg': 26, 'aut_perms': [15564072393114374965223069, 400420762366171630674078115, 47593564096117594345908966, 325372279284144463838654205, 402668125760929192322280783, 15720066233386811395571914, 1758207700455073377402844], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 14, 1, 1], [2, 18, 1, 1], [2, 42, 1, 1], [2, 63, 2, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 6, 1, 1], [4, 42, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 12, 1, 1], [6, 28, 1, 2], [6, 28, 2, 1], [6, 84, 1, 1], [7, 2, 3, 1], [12, 12, 1, 1], [12, 84, 1, 1], [14, 2, 3, 1], [14, 12, 3, 1], [14, 36, 3, 1], [21, 4, 3, 2], [21, 8, 3, 1], [28, 12, 3, 1], [42, 4, 3, 2], [42, 8, 3, 1], [42, 12, 6, 1], [84, 12, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3\\times S_3^2\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '1512.792', 'autcentquo_hash': 792, 'autcentquo_nilpotent': False, 'autcentquo_order': 1512, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 1], [2, 14, 1], [2, 18, 1], [2, 42, 1], [2, 63, 2], [3, 2, 2], [3, 4, 1], [4, 6, 1], [4, 42, 1], [6, 2, 2], [6, 4, 1], [6, 12, 1], [6, 28, 4], [6, 84, 1], [7, 2, 3], [12, 12, 1], [12, 84, 1], [14, 2, 3], [14, 12, 3], [14, 36, 3], [21, 4, 6], [21, 8, 3], [28, 12, 3], [42, 4, 6], [42, 8, 3], [42, 12, 6], [84, 12, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '504.172', 'commutator_count': 1, 'commutator_label': '126.16', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 598, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 14, 1, 1], [2, 18, 1, 1], [2, 42, 1, 1], [2, 63, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [4, 6, 1, 1], [4, 42, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 12, 1, 1], [6, 28, 1, 4], [6, 84, 1, 1], [7, 2, 3, 1], [12, 12, 1, 1], [12, 84, 1, 1], [14, 2, 3, 1], [14, 12, 3, 1], [14, 36, 3, 1], [21, 4, 3, 2], [21, 8, 3, 1], [28, 12, 3, 1], [42, 4, 3, 2], [42, 8, 3, 1], [42, 12, 6, 1], [84, 12, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 21504, 'exponent': 84, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, 1, 3]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '504.172', 'hash': 598, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 2, 42, 3], 'inner_gens': [[1, 170, 284, 336], [169, 2, 220, 336], [57, 122, 4, 672], [1, 2, 676, 336]], 'inner_hash': 172, 'inner_nilpotent': False, 'inner_order': 504, 'inner_split': True, 'inner_tex': 'D_7\\times S_3^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 22], [4, 33], [8, 6]], 'label': '1008.598', 'linC_count': 72, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 8, 'linQ_dim': 10, 'linQ_dim_count': 8, 'linR_count': 48, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D42:D6', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 35, 'number_characteristic_subgroups': 46, 'number_conjugacy_classes': 69, 'number_divisions': 37, 'number_normal_subgroups': 48, 'number_subgroup_autclasses': 212, 'number_subgroup_classes': 248, 'number_subgroups': 3212, 'old_label': None, 'order': 1008, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 207], [3, 8], [4, 48], [6, 216], [7, 6], [12, 96], [14, 150], [21, 48], [28, 36], [42, 120], [84, 72]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 2, 116, 336], [1, 2, 116, 672], [1, 170, 244, 672]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [744, 24, 40323], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 6], [6, 4], [12, 5], [24, 4]], 'representations': {'PC': {'code': 41625038372093394860716914416222930747326216771, 'gens': [1, 2, 3, 7], 'pres': [7, -2, -2, -2, -2, -3, -7, -3, 2381, 5966, 2319, 58, 6499, 1466, 80, 4484, 3651, 137, 6060, 8252]}, 'Perm': {'d': 17, 'gens': [1313941305720, 41064, 1465, 2424, 403200, 3, 23550632582400]}}, 'schur_multiplier': [2, 2, 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': 'D_{42}:D_6', 'transitive_degree': 84, 'wreath_data': None, 'wreath_product': False}
