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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '88.12', 'ambient_counter': 12, 'ambient_order': 88, 'ambient_tex': 'C_2^2\\times C_{22}', 'central': True, 'central_factor': False, 'centralizer_order': 88, 'characteristic': False, 'core_order': 2, 'counter': 25, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '88.12.44.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '44.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '44.4', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 44, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_{22}', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '88.12', 'aut_centralizer_order': 240, 'aut_label': '44.a1', 'aut_quo_index': 1, 'aut_stab_index': 7, 'aut_weyl_group': '1.1', 'aut_weyl_index': 1680, 'centralizer': '1.a1.a1', 'complements': ['2.a1.d1', '2.a1.e1', '2.a1.f1', '2.a1.g1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '22.a1.a1', '22.a1.b1', '22.a1.c1'], 'contains': ['88.a1.a1'], 'core': '44.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4049, 4049, 5419, 5419, 3297, 3297, 8454, 8454], 'generators': [44], 'label': '88.12.44.a1.a1', 'mobius_quo': -1, 'mobius_sub': -2, 'normal_closure': '44.a1.a1', 'normal_contained_in': ['4.a1.a1', '22.a1.b1', '22.a1.a1', '22.a1.c1'], 'normal_contains': ['88.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '44.a1.a1', 'projective_image': '44.4', 'quotient_action_image': '1.1', 'quotient_action_kernel': '44.4', 'quotient_action_kernel_order': 44, 'quotient_fusion': None, 'short_label': '44.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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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}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '88.12', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 420, 'aut_gen_orders': [10, 10, 5, 10, 2, 2], 'aut_gens': [[1, 2, 4], [2, 1, 79], [47, 44, 82], [1, 2, 36], [1, 2, 68], [3, 2, 4], [1, 2, 86]], 'aut_group': '1680.928', 'aut_hash': 928, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1680, 'aut_permdeg': 14, 'aut_perms': [25514095321, 31741477350, 870, 2881, 6234641280, 171319681], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [11, 1, 10, 1], [22, 1, 70, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{10}\\times \\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 420, 'autcent_group': '1680.928', 'autcent_hash': 928, 'autcent_nilpotent': False, 'autcent_order': 1680, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_{10}\\times \\GL(3,2)', '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, 7], [11, 1, 10], [22, 1, 70]], 'center_label': '88.12', 'center_order': 88, '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', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [11, 1, 10, 1], [22, 1, 10, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 133, 'exponent': 22, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '88.12', 'hash': 12, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 88]], 'label': '88.12', 'linC_count': 37240, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 84, 'linQ_dim': 12, 'linQ_dim_count': 84, 'linR_count': 420, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C22', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 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': 88, 'number_divisions': 16, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 32, 'number_subgroups': 32, 'old_label': None, 'order': 88, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [11, 10], [22, 70]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 420, 'outer_gen_orders': [10, 10, 5, 10, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[2, 1, 79], [47, 44, 82], [1, 2, 36], [1, 2, 68], [3, 2, 4], [1, 2, 86]], 'outer_group': '1680.928', 'outer_hash': 928, 'outer_nilpotent': False, 'outer_order': 1680, 'outer_permdeg': 14, 'outer_perms': [25514095321, 31741477350, 870, 2881, 6234641280, 171319681], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_{10}\\times \\GL(3,2)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 11], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [10, 8]], 'representations': {'PC': {'code': 5160009, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -11, 34]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [826574, 359380, 36037, 35960]}, 'Perm': {'d': 17, 'gens': [20922789888000, 87178291200, 479001600, 36288000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 22], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{22}', 'transitive_degree': 88, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '44.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 30, 'aut_gen_orders': [2, 30], 'aut_gens': [[1, 2], [22, 21], [23, 37]], 'aut_group': '60.11', 'aut_hash': 11, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 60, 'aut_permdeg': 10, 'aut_perms': [1, 816508], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [11, 1, 10, 1], [22, 1, 30, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times C_{10}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 30, 'autcent_group': '60.11', 'autcent_hash': 11, 'autcent_nilpotent': False, 'autcent_order': 60, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3\\times C_{10}', '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], [11, 1, 10], [22, 1, 30]], 'center_label': '44.4', 'center_order': 44, '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', '11.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [11, 1, 10, 1], [22, 1, 10, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 22, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '44.4', 'hash': 4, '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, 44]], 'label': '44.4', 'linC_count': 360, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 6, 'linQ_dim': 11, 'linQ_dim_count': 6, 'linR_count': 30, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C22', 'ngens': 3, 'nilpotency_class': 1, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 44, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 44, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [11, 10], [22, 30]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 30], 'outer_gen_pows': [0, 0], 'outer_gens': [[22, 21], [23, 37]], 'outer_group': '60.11', 'outer_hash': 11, 'outer_nilpotent': False, 'outer_order': 60, 'outer_permdeg': 10, 'outer_perms': [1, 816508], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times C_{10}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [10, 4]], 'representations': {'PC': {'code': 34009, 'gens': [1, 2], 'pres': [3, -2, -2, -11, 16]}, 'GLFp': {'d': 2, 'p': 23, 'gens': [146014, 267696]}, 'Perm': {'d': 15, 'gens': [87178291200, 479001600, 36288000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 22], '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_{22}', 'transitive_degree': 44, 'wreath_data': None, 'wreath_product': False}
