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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '180.17', 'ambient_counter': 17, 'ambient_order': 180, 'ambient_tex': 'C_6.D_{15}', 'central': False, 'central_factor': False, 'centralizer_order': 90, 'characteristic': False, 'core_order': 15, 'counter': 18, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '180.17.12.a1.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '12.a1.d1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '12.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 12, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3:C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '15.1', 'subgroup_hash': 1, 'subgroup_order': 15, 'subgroup_tex': 'C_{15}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '180.17', 'aut_centralizer_order': 540, 'aut_label': '12.a1', 'aut_quo_index': 1, 'aut_stab_index': 4, 'aut_weyl_group': '8.2', 'aut_weyl_index': 2160, 'centralizer': '2.a1.a1', 'complements': ['15.a1.a1', '15.a1.b1', '15.a1.c1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '6.a1.d1'], 'contains': ['36.a1.a1', '60.a1.d1'], 'core': '12.a1.d1', 'coset_action_label': None, 'count': 1, 'diagramx': [6674, 3507, 7372, 9678, 4598, 6676, 1884, 1553], 'generators': [64, 36], 'label': '180.17.12.a1.d1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '12.a1.d1', 'normal_contained_in': ['4.a1.a1', '6.a1.d1'], 'normal_contains': ['36.a1.a1', '60.a1.d1'], 'normalizer': '1.a1.a1', 'old_label': '12.a1.d1', 'projective_image': '180.17', 'quotient_action_image': '2.1', 'quotient_action_kernel': '6.2', 'quotient_action_kernel_order': 6, 'quotient_fusion': None, 'short_label': '12.a1.d1', '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': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1], [11], [7]], 'aut_group': '8.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 6, 'aut_perms': [120, 9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [5, 1, 4, 1], [15, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_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], [3, 1, 2], [5, 1, 4], [15, 1, 8]], 'center_label': '15.1', 'center_order': 15, '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': ['3.1', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [5, 1, 4, 1], [15, 1, 8, 1]], 'element_repr_type': 'PC', 'elementary': 15, 'eulerian_function': 1, 'exponent': 15, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [3, 5], 'faithful_reps': [[1, 0, 8]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '15.1', 'hash': 1, 'hyperelementary': 15, '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': 8, 'irrQ_dim': 8, 'irrR_degree': 2, 'irrep_stats': [[1, 15]], 'label': '15.1', 'linC_count': 8, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 4, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C15', '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': 15, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 15, 'order_factorization_type': 11, 'order_stats': [[1, 1], [3, 2], [5, 4], [15, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[11], [7]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [4, 1], [8, 1]], 'representations': {'PC': {'code': 87, 'gens': [1], 'pres': [2, -3, -5, 6]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58415890696506688]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [6351]}, 'Perm': {'d': 8, 'gens': [10080, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}', 'transitive_degree': 15, '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': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [2, 6, 2, 4, 6, 6, 5], 'aut_gens': [[1, 4, 12], [3, 4, 168], [61, 4, 48], [3, 4, 48], [1, 4, 156], [9, 4, 48], [1, 120, 176], [145, 4, 12]], 'aut_group': '17280.fp', 'aut_hash': 7336759155281172877, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 17280, 'aut_permdeg': 16, 'aut_perms': [25517722014, 8371514592414, 415, 169, 3941704892574, 126504000414, 2928], 'aut_phi_ratio': 360.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 4, 1], [4, 45, 2, 1], [5, 2, 2, 1], [6, 2, 4, 1], [10, 2, 2, 1], [15, 2, 16, 1], [30, 2, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{30}:C_{12}:\\GL(2,3)', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '8640.be', 'autcentquo_hash': 9203803121985669655, 'autcentquo_nilpotent': False, 'autcentquo_order': 8640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 4], [4, 45, 2], [5, 2, 2], [6, 2, 4], [10, 2, 2], [15, 2, 16], [30, 2, 16]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '90.9', 'commutator_count': 1, 'commutator_label': '45.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 17, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 4], [4, 45, 2, 1], [5, 2, 2, 1], [6, 2, 1, 4], [10, 2, 2, 1], [15, 2, 4, 4], [30, 2, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 168, 'exponent': 60, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '90.9', 'hash': 17, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 3, 15], 'inner_gens': [[1, 8, 168], [9, 4, 12], [25, 4, 12]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 90, 'inner_split': False, 'inner_tex': 'C_3:D_{15}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 44]], 'label': '180.17', 'linC_count': 432, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 42, 'linQ_dim': 10, 'linQ_dim_count': 18, 'linR_count': 432, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6.D15', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 48, 'number_divisions': 21, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 36, 'number_subgroups': 156, 'old_label': None, 'order': 180, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 90], [5, 4], [6, 8], [10, 4], [15, 32], [30, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 6, 4, 2, 4], 'outer_gen_pows': [0, 0, 0, 1, 0], 'outer_gens': [[3, 8, 48], [3, 128, 132], [1, 124, 136], [3, 68, 164], [1, 64, 140]], 'outer_group': '192.1480', 'outer_hash': 1480, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 18, 'outer_perms': [359903136438864, 4278468223105231, 3026838283310190, 4914020596729320, 1578989323193088], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3):C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 2], [8, 8]], 'representations': {'PC': {'code': 53724910819976215, 'gens': [1, 3, 4], 'pres': [5, -2, -2, -3, -3, -5, 10, 122, 3363, 78, 3604]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [27301, 40966, 203123, 27181, 513019]}, 'Perm': {'d': 15, 'gens': [6267345865, 806400, 3, 148, 99672249600]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6.D_{15}', 'transitive_degree': 180, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 4], [1, 8], [11, 4]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 3, 2, 1], [6, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 1], [4, 3, 2], [6, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 3, 2, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 12, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, -1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 8], [9, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.1', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3:C4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 6, 'number_divisions': 5, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 8, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 6], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3, 4]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3]], 'representations': {'PC': {'code': 3913, 'gens': [1, 3], 'pres': [3, -2, -2, -3, 6, 74]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20468488, 16858410]}, 'Lie': [{'d': 2, 'q': 7, 'gens': [2064, 1718, 690, 273], 'family': 'CSOPlus'}, {'d': 2, 'q': 5, 'gens': [537, 504, 427], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [169, 458]}, 'Perm': {'d': 7, 'gens': [129, 16, 840]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:C_4', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
