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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '500.5', 'ambient_counter': 5, 'ambient_order': 500, 'ambient_tex': 'C_2\\times C_{250}', 'central': True, 'central_factor': False, 'centralizer_order': 500, 'characteristic': True, 'core_order': 1, 'counter': 20, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '500.5.500.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '500.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '500.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 500, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_{250}', 'simple': False, 'solvable': True, 'special_labels': ['D', 'PC', 'U1', 'L1', 'D1', 'C5'], 'split': True, 'standard_generators': False, 'stem': True, 'subgroup': '1.1', 'subgroup_hash': 1, 'subgroup_order': 1, 'subgroup_tex': 'C_1', 'supersolvable': True, 'sylow': 1}
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gps_subgroup_data • Show schema
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{'ambient': '500.5', 'aut_centralizer_order': None, 'aut_label': '500.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1.a1', 'complements': ['1.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['100.a1.a1', '250.a1.a1', '250.a1.b1', '250.a1.c1'], 'contains': [], 'core': '500.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5875, 5875, 6359, 6359, 5242, 5242, 6984, 6984], 'generators': [], 'label': '500.5.500.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '500.a1.a1', 'normal_contained_in': ['100.a1.a1', '250.a1.b1', '250.a1.c1', '250.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '500.a1.a1', 'projective_image': '500.5', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '500.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': '1.1', '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': [], '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]], '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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 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': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, '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': '500.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 300, 'aut_gen_orders': [2, 300], 'aut_gens': [[1, 2], [251, 2], [250, 395]], 'aut_group': '600.21', 'aut_hash': 21, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 600, 'aut_permdeg': 103, 'aut_perms': [59619213849979307046865777122188699414521500497216203317346343098721688940353324318388054124862281393352485560881574727947126684875329489777915298280636416000000000, 59561292878738794786937848194200546193907750721649279448566517448055120469030219590465042534959548425531520493709805503995476339602716770707857113753244162152602874], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 4, 1], [10, 1, 12, 1], [25, 1, 20, 1], [50, 1, 60, 1], [125, 1, 100, 1], [250, 1, 300, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times C_{100}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 300, 'autcent_group': '600.21', 'autcent_hash': 21, 'autcent_nilpotent': False, 'autcent_order': 600, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3\\times C_{100}', '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], [5, 1, 4], [10, 1, 12], [25, 1, 20], [50, 1, 60], [125, 1, 100], [250, 1, 300]], 'center_label': '500.5', 'center_order': 500, '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', '5.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['125.1', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 1], [10, 1, 4, 3], [25, 1, 20, 1], [50, 1, 20, 3], [125, 1, 100, 1], [250, 1, 100, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 150, 'exponent': 250, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '25.1', 'frattini_quotient': '20.5', 'hash': 5, '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, 500]], 'label': '500.5', 'linC_count': 45000, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 101, 'linQ_degree_count': 6, 'linQ_dim': 101, 'linQ_dim_count': 6, 'linR_count': 300, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C250', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 500, 'number_divisions': 16, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 20, 'number_subgroups': 20, 'old_label': None, 'order': 500, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [5, 4], [10, 12], [25, 20], [50, 60], [125, 100], [250, 300]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 300, 'outer_gen_orders': [2, 300], 'outer_gen_pows': [0, 0], 'outer_gens': [[251, 2], [250, 395]], 'outer_group': '600.21', 'outer_hash': 21, 'outer_nilpotent': False, 'outer_order': 600, 'outer_permdeg': 103, 'outer_perms': [59619213849979307046865777122188699414521500497216203317346343098721688940353324318388054124862281393352485560881574727947126684875329489777915298280636416000000000, 59561292878738794786937848194200546193907750721649279448566517448055120469030219590465042534959548425531520493709805503995476339602716770707857113753244162152602874], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times C_{100}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 129, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 125], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4], [20, 4], [100, 4]], 'representations': {'PC': {'code': 4236666484799, 'gens': [1, 2], 'pres': [5, -2, -2, -5, -5, -5, 26, 87, 118]}, 'GLFp': {'d': 2, 'p': 251, 'gens': [1043674703, 3953313000]}, 'Perm': {'d': 129, 'gens': [385620482362580421735677065923463640617493109590223590278828403276373402575165543560686168588507361534030051833058916347592172932262498857766114955245039357760034644709279247692495585280000000000000000000000000000000, 23721732428800468856771473051394170805702085973808045661837377170052497697783313457227249544076486314839447086187187275319400401837013955325179315652376928996065123321190898603130880000000000000000000000000000000, 188231080464409993308566180397956207174011130878381467672701527747899312789095433103790426415702341803211489914270009794686047389651890638822766658702240221929735288630664826883512729600000000000000000000000000, 36392319214835163244738167583658704005052458954624684534212778504466968576171427322617887977935194857673802640600640284362242453306352079924561004347542145493434531341187049286465286303136937699932355952640000, 6024566966267546949638535122160526306836773483629472782180425807322809231718001870781827628377008839797943269016650950164203401811756882442005046860365939897473474062078934525650278250855413970146955851384416]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 250], '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_{250}', 'transitive_degree': 500, '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': '500.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 300, 'aut_gen_orders': [2, 300], 'aut_gens': [[1, 2], [251, 2], [250, 395]], 'aut_group': '600.21', 'aut_hash': 21, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 600, 'aut_permdeg': 103, 'aut_perms': [59619213849979307046865777122188699414521500497216203317346343098721688940353324318388054124862281393352485560881574727947126684875329489777915298280636416000000000, 59561292878738794786937848194200546193907750721649279448566517448055120469030219590465042534959548425531520493709805503995476339602716770707857113753244162152602874], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 4, 1], [10, 1, 12, 1], [25, 1, 20, 1], [50, 1, 60, 1], [125, 1, 100, 1], [250, 1, 300, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times C_{100}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 300, 'autcent_group': '600.21', 'autcent_hash': 21, 'autcent_nilpotent': False, 'autcent_order': 600, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3\\times C_{100}', '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], [5, 1, 4], [10, 1, 12], [25, 1, 20], [50, 1, 60], [125, 1, 100], [250, 1, 300]], 'center_label': '500.5', 'center_order': 500, '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', '5.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['125.1', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 1], [10, 1, 4, 3], [25, 1, 20, 1], [50, 1, 20, 3], [125, 1, 100, 1], [250, 1, 100, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 150, 'exponent': 250, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '25.1', 'frattini_quotient': '20.5', 'hash': 5, '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, 500]], 'label': '500.5', 'linC_count': 45000, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 101, 'linQ_degree_count': 6, 'linQ_dim': 101, 'linQ_dim_count': 6, 'linR_count': 300, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C250', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 500, 'number_divisions': 16, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 20, 'number_subgroups': 20, 'old_label': None, 'order': 500, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [5, 4], [10, 12], [25, 20], [50, 60], [125, 100], [250, 300]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 300, 'outer_gen_orders': [2, 300], 'outer_gen_pows': [0, 0], 'outer_gens': [[251, 2], [250, 395]], 'outer_group': '600.21', 'outer_hash': 21, 'outer_nilpotent': False, 'outer_order': 600, 'outer_permdeg': 103, 'outer_perms': [59619213849979307046865777122188699414521500497216203317346343098721688940353324318388054124862281393352485560881574727947126684875329489777915298280636416000000000, 59561292878738794786937848194200546193907750721649279448566517448055120469030219590465042534959548425531520493709805503995476339602716770707857113753244162152602874], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times C_{100}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 129, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 125], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4], [20, 4], [100, 4]], 'representations': {'PC': {'code': 4236666484799, 'gens': [1, 2], 'pres': [5, -2, -2, -5, -5, -5, 26, 87, 118]}, 'GLFp': {'d': 2, 'p': 251, 'gens': [1043674703, 3953313000]}, 'Perm': {'d': 129, 'gens': [385620482362580421735677065923463640617493109590223590278828403276373402575165543560686168588507361534030051833058916347592172932262498857766114955245039357760034644709279247692495585280000000000000000000000000000000, 23721732428800468856771473051394170805702085973808045661837377170052497697783313457227249544076486314839447086187187275319400401837013955325179315652376928996065123321190898603130880000000000000000000000000000000, 188231080464409993308566180397956207174011130878381467672701527747899312789095433103790426415702341803211489914270009794686047389651890638822766658702240221929735288630664826883512729600000000000000000000000000, 36392319214835163244738167583658704005052458954624684534212778504466968576171427322617887977935194857673802640600640284362242453306352079924561004347542145493434531341187049286465286303136937699932355952640000, 6024566966267546949638535122160526306836773483629472782180425807322809231718001870781827628377008839797943269016650950164203401811756882442005046860365939897473474062078934525650278250855413970146955851384416]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 250], '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_{250}', 'transitive_degree': 500, 'wreath_data': None, 'wreath_product': False}
