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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '192.74', 'ambient_counter': 74, 'ambient_order': 192, 'ambient_tex': 'C_8.D_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 16, 'counter': 14, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '192.74.6.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '6.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': 6, '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': '32.38', 'subgroup_hash': 38, 'subgroup_order': 32, 'subgroup_tex': '\\OD_{16}:C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '192.74', 'aut_centralizer_order': 4, 'aut_label': '6.b1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '32.46', 'aut_weyl_index': 12, 'centralizer': '24.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '3.a1.a1'], 'contains': ['12.a1.a1', '12.b1.a1', '12.c1.a1', '12.d1.a1', '12.e1.a1'], 'core': '12.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [2538, -1, 7655, -1, 6044, -1, 6044, -1], 'generators': [1, 4, 144], 'label': '192.74.6.b1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '6.b1.a1', 'projective_image': '24.8', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.b1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 6, 2, 2], 'aut_gens': [[1722, 1860894, 620321], [68880, 2619036, 620321], [2756841, 2619036, 15498], [68961, 206766, 3321], [68880, 1860894, 620321], [68880, 1860894, 2205481]], 'aut_group': '96.226', 'aut_hash': 226, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [7, 120, 856, 16680, 5160], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [4, 1, 2, 1], [4, 2, 3, 1], [8, 1, 4, 1], [8, 2, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', '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], [2, 2, 3], [4, 1, 2], [4, 2, 3], [8, 1, 4], [8, 2, 6]], 'center_label': '8.1', 'center_order': 8, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 38, '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, 3], [4, 1, 2, 1], [4, 2, 1, 3], [8, 1, 4, 1], [8, 2, 2, 3]], 'element_repr_type': 'GLFp', 'elementary': 2, 'eulerian_function': 112, 'exponent': 8, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '8.5', 'hash': 38, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 1, 2], 'inner_gens': [[1722, 1860894, 2205481], [1722, 1860894, 620321], [68880, 1860894, 620321]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 3], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 16], [2, 4]], 'label': '32.38', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'OD16:C2', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 20, 'number_divisions': 13, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 15, 'number_subgroup_classes': 31, 'number_subgroups': 34, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 8], [8, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [68922, 620321, 68922], 'outer_gens': [[1722, 206766, 2205481], [54161, 1860894, 620321], [68961, 964908, 67281]], 'outer_group': '24.14', 'outer_hash': 14, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [16, 127, 847], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [8, 1]], 'representations': {'PC': {'code': 17334300, 'gens': [1, 2, 3], 'pres': [5, -2, 2, 2, -2, -2, 181, 42, 58]}, 'GLFp': {'d': 2, 'p': 41, 'gens': [1722, 1860894, 620321]}, 'Perm': {'d': 16, 'gens': [15310327737600, 8402294108878, 5606234750016, 4097506710982, 1313941673647]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\OD_{16}:C_2', 'transitive_degree': 16, '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': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 4, 12], 'aut_gens': [[1, 2, 16], [145, 110, 112], [1, 110, 16], [1, 2, 80], [1, 10, 16], [113, 146, 16]], 'aut_group': '384.18344', 'aut_hash': 18344, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 13, 'aut_perms': [288, 367921, 39916927, 811440, 518918808], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 1, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 12, 1, 1], [6, 2, 1, 1], [6, 4, 1, 1], [8, 1, 4, 1], [8, 2, 2, 1], [8, 12, 2, 1], [12, 2, 2, 1], [12, 4, 1, 1], [16, 4, 4, 1], [16, 6, 8, 1], [24, 2, 4, 1], [24, 4, 2, 1], [48, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times D_4\\times D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '24.14', 'autcentquo_hash': 14, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 12, 1], [3, 2, 1], [4, 1, 2], [4, 2, 1], [4, 12, 1], [6, 2, 1], [6, 4, 1], [8, 1, 4], [8, 2, 2], [8, 12, 2], [12, 2, 2], [12, 4, 1], [16, 4, 4], [16, 6, 8], [24, 2, 4], [24, 4, 2], [48, 4, 8]], 'center_label': '8.1', 'center_order': 8, 'central_product': False, 'central_quotient': '24.8', 'commutator_count': 1, 'commutator_label': '12.2', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 74, '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, 12, 1, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 12, 1, 1], [6, 2, 1, 1], [6, 4, 1, 1], [8, 1, 4, 1], [8, 2, 2, 1], [8, 12, 2, 1], [12, 2, 2, 1], [12, 4, 1, 1], [16, 4, 4, 1], [16, 6, 8, 1], [24, 2, 4, 1], [24, 4, 2, 1], [48, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 48, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 4]], 'familial': False, 'frattini_label': '16.5', 'frattini_quotient': '12.4', 'hash': 74, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 6], 'inner_gens': [[1, 146, 176], [49, 2, 112], [33, 98, 16]], 'inner_hash': 8, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_3:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 32, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 28], [4, 4]], 'label': '192.74', 'linC_count': 132, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 18, 'linQ_dim_count': 2, 'linR_count': 16, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C8.D12', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 20, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 48, 'number_divisions': 20, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 56, 'number_subgroups': 136, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 15], [3, 2], [4, 16], [6, 6], [8, 32], [12, 8], [16, 64], [24, 16], [48, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 146, 112], [1, 14, 16], [1, 6, 16]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [120, 5040, 5057], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 35, 'pgroup': 0, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 6], [8, 2], [16, 2]], 'representations': {'PC': {'code': 24398435671700685447714289696513, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -3, 2045, 36, 58, 360, 6164, 1971, 102, 6725, 124, 6278]}, 'GLZN': {'d': 2, 'p': 34, 'gens': [510965, 353745, 1297065, 699261, 39337, 3842, 727703]}, 'Perm': {'d': 35, 'gens': [297654082100529507678254198482017465361, 643169832186062607479825574027124378800, 949148838713262223333685213634105613464, 1253392177629462570167035065181528533120, 1548631908697581146324001321859049677464, 1852820704392598329905848752888032819784, 3]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_8.D_{12}', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
