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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '784.135', 'ambient_counter': 135, 'ambient_order': 784, 'ambient_tex': 'D_{14}:D_{14}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': True, 'core_order': 392, 'counter': 6, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '784.135.2.e1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.e1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '392.19', 'subgroup_hash': 19, 'subgroup_order': 392, 'subgroup_tex': 'C_{14}.D_{14}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '784.135', 'aut_centralizer_order': 2, 'aut_label': '2.e1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '14112.ba', 'aut_weyl_index': 2, 'centralizer': '392.a1', 'complements': ['392.b1', '392.c1', '392.e1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1'], 'contains': ['4.c1', '4.e1', '14.h1'], 'core': '2.e1', 'coset_action_label': None, 'count': 1, 'diagramx': [3920, 2370, 7377, 499], 'generators': [29, 392, 8, 398, 112], 'label': '784.135.2.e1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.e1', 'normal_contained_in': ['1.a1'], 'normal_contains': ['4.c1', '4.e1'], 'normalizer': '1.a1', 'old_label': '2.e1', 'projective_image': '392.41', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.e1', 'subgroup_fusion': None, 'weyl_group': '392.41'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [6, 4, 6, 14, 7], 'aut_gens': [[1, 2, 56], [14, 141, 32], [294, 129, 8], [53, 286, 280], [1, 390, 56], [41, 2, 56]], 'aut_group': '14112.ba', 'aut_hash': 3417587930551382414, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14112, 'aut_permdeg': 28, 'aut_perms': [224384298778281486508125837735, 78048480572235104507773615920, 168948637811527502233634877830, 131423430069942258994201665399, 135140567919338826283867033113], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 49, 2, 1], [4, 7, 4, 1], [7, 2, 6, 1], [7, 4, 9, 1], [14, 2, 6, 1], [14, 4, 9, 1], [28, 14, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_7^2:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '3528.br', 'autcentquo_hash': 519392570028930314, 'autcentquo_nilpotent': False, 'autcentquo_order': 3528, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_7\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 49, 2], [4, 7, 4], [7, 2, 6], [7, 4, 9], [14, 2, 6], [14, 4, 9], [28, 14, 12]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '196.9', 'commutator_count': 1, 'commutator_label': '49.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '7.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 19, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 49, 1, 2], [4, 7, 2, 2], [7, 2, 3, 2], [7, 4, 3, 3], [14, 2, 3, 2], [14, 4, 3, 3], [28, 14, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 28, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 1, 9]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '196.9', 'hash': 19, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 14, 7], 'inner_gens': [[1, 26, 56], [33, 2, 336], [1, 114, 56]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 196, 'inner_split': True, 'inner_tex': 'D_7^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 24], [4, 18]], 'label': '392.19', 'linC_count': 117, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 3, 'linQ_dim': 12, 'linQ_dim_count': 3, 'linR_count': 9, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C14.D14', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 50, 'number_divisions': 18, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 23, 'number_subgroup_classes': 47, 'number_subgroups': 434, 'old_label': None, 'order': 392, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 99], [4, 28], [7, 48], [14, 48], [28, 168]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [6, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[29, 46, 168], [14, 85, 16], [1, 46, 56]], 'outer_group': '72.30', 'outer_hash': 30, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 10, 'outer_perms': [458663, 1629362, 448576], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\wr C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [6, 4], [12, 8]], 'representations': {'PC': {'code': 2352733734202412050652867, 'gens': [1, 2, 5], 'pres': [5, -2, -2, -2, -7, -7, 140, 261, 26, 782, 42, 963, 4209]}, 'GLZN': {'d': 2, 'p': 91, 'gens': [6030968, 16927702, 754755, 6030536, 48228608]}, 'Perm': {'d': 18, 'gens': [4354687, 21010454478720, 11652480, 397707843456000, 973]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{14}.D_{14}', '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': 3, 'aut_exponent': 84, 'aut_gen_orders': [4, 6, 12, 6, 6], 'aut_gens': [[1, 2, 4, 56], [426, 57, 700, 408], [729, 442, 12, 168], [10, 617, 532, 416], [1, 42, 412, 728], [617, 18, 436, 168]], 'aut_group': None, 'aut_hash': 1340277925156936142, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 28224, 'aut_permdeg': 30, 'aut_perms': [7387007501702321841681664172549, 2908053171872770874609011908148, 243048555513882520302747422509795, 245739440436090054431848785677245, 240188197248611969552284632953460], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 14, 2, 1], [2, 49, 2, 1], [2, 98, 1, 1], [4, 14, 2, 1], [7, 2, 6, 1], [7, 4, 9, 1], [14, 2, 6, 1], [14, 4, 6, 1], [14, 4, 9, 1], [14, 4, 18, 1], [14, 28, 6, 1], [28, 28, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_7^2.C_6^2.C_2^4', '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': 84, 'autcentquo_group': '3528.br', 'autcentquo_hash': 519392570028930314, 'autcentquo_nilpotent': False, 'autcentquo_order': 3528, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_7\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 14, 2], [2, 49, 2], [2, 98, 1], [4, 14, 2], [7, 2, 6], [7, 4, 9], [14, 2, 6], [14, 4, 33], [14, 28, 6], [28, 28, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '392.41', 'commutator_count': 1, 'commutator_label': '98.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '7.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 135, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 14, 1, 2], [2, 49, 1, 2], [2, 98, 1, 1], [4, 14, 1, 2], [7, 2, 3, 2], [7, 4, 3, 3], [14, 2, 3, 2], [14, 4, 3, 11], [14, 28, 3, 2], [28, 28, 3, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5376, 'exponent': 28, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 1, 18]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '392.41', 'hash': 135, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 2, 14, 7], 'inner_gens': [[1, 2, 396, 728], [1, 2, 444, 56], [393, 402, 4, 56], [113, 2, 4, 56]], 'inner_hash': 41, 'inner_nilpotent': False, 'inner_order': 392, 'inner_split': False, 'inner_tex': 'D_7\\times D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 26], [4, 42]], 'label': '784.135', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'D14:D14', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 15, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 76, 'number_divisions': 32, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 156, 'number_subgroups': 1952, 'old_label': None, 'order': 784, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 227], [4, 28], [7, 48], [14, 312], [28, 168]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 394, 4, 728], [2, 1, 140, 424], [1, 2, 428, 616]], 'outer_group': '72.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 10, 'outer_perms': [367920, 145, 806404], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, '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, 8], [12, 14]], 'representations': {'PC': {'code': 796608832822951082209472777657, 'gens': [1, 2, 3, 5], 'pres': [6, -2, -2, -2, -7, -2, -7, 7130, 4004, 50, 585, 21844, 88, 24197]}, 'Perm': {'d': 18, 'gens': [21010447946880, 766207, 3669120, 7983360, 973, 397707843456000]}}, '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_{14}:D_{14}', 'transitive_degree': 28, 'wreath_data': None, 'wreath_product': False}
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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}
