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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '900.131', 'ambient_counter': 131, 'ambient_order': 900, 'ambient_tex': 'C_{15}^2:C_2^2', 'central': True, 'central_factor': False, 'centralizer_order': 900, 'characteristic': True, 'core_order': 1, 'counter': 45, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '900.131.900.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '900.a1', 'outer_equivalence': True, 'perfect': True, 'proper': False, 'quotient': '900.131', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 131, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 900, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{15}^2:C_2^2', 'simple': False, 'solvable': True, 'special_labels': ['Z', 'Phi', 'PC', 'U0', 'D2', 'C6'], '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': '900.131', 'aut_centralizer_order': None, 'aut_label': '900.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1', 'complements': ['1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['180.a1', '180.b1', '300.a1', '450.a1', '450.b1'], 'contains': [], 'core': '900.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2220, 6176, 7295, 3933], 'generators': [], 'label': '900.131.900.a1', 'mobius_quo': 1, 'mobius_sub': 1350, 'normal_closure': '900.a1', 'normal_contained_in': ['180.a1', '300.a1'], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '900.a1', 'projective_image': '900.131', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '900.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': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [12, 12, 8, 60], 'aut_gens': [[1, 2, 4, 60], [733, 122, 656, 780], [361, 343, 880, 324], [541, 282, 328, 740], [757, 142, 44, 780]], 'aut_group': None, 'aut_hash': 2436627162132881625, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 345600, 'aut_permdeg': 90, 'aut_perms': [1317905993949449910595619579072711277353925821947402462258111618689098790614008122274708605306091730288780453792046726394595432097576765148, 1269567077681077699560137000948903663775642455131507690020963826412852985773859619641014997948047447771572393999517114689740433731455492806, 411907624817898892856726234153072531424231865265091162739970789103174503127107609169219617767980292552542683555021161260427483622621627787, 113116181324502536131239890590033757356086697392372493528661673651499081716327326725139190557286505778793158098015797064719646117818810003], 'aut_phi_ratio': 1440.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [2, 45, 2, 1], [3, 2, 4, 1], [5, 2, 4, 1], [5, 4, 4, 1], [6, 50, 4, 1], [10, 90, 4, 1], [15, 4, 16, 1], [15, 4, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3:S_3.C_{10}^2.C_{12}.C_4.C_2^2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': None, 'autcentquo_hash': 2436627162132881625, 'autcentquo_nilpotent': False, 'autcentquo_order': 345600, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3:S_3.C_{10}^2.C_{12}.C_4.C_2^2', 'cc_stats': [[1, 1, 1], [2, 25, 1], [2, 45, 2], [3, 2, 4], [5, 2, 4], [5, 4, 4], [6, 50, 4], [10, 90, 4], [15, 4, 48]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '900.131', 'commutator_count': 1, 'commutator_label': '225.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 131, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [2, 45, 1, 2], [3, 2, 1, 4], [5, 2, 2, 2], [5, 4, 2, 2], [6, 50, 1, 4], [10, 90, 2, 2], [15, 4, 2, 8], [15, 4, 4, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 756, 'exponent': 30, 'exponents_of_order': [2, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '900.131', 'hash': 131, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 2, 15, 15], 'inner_gens': [[1, 2, 16, 240], [1, 2, 44, 840], [49, 22, 4, 60], [721, 122, 4, 60]], 'inner_hash': 131, 'inner_nilpotent': False, 'inner_order': 900, 'inner_split': False, 'inner_tex': 'C_{15}^2:C_2^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 16], [4, 52]], 'label': '900.131', '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': 'C15^2:C2^2', 'ngens': 6, '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, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 72, 'number_divisions': 34, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 45, 'number_subgroup_classes': 144, 'number_subgroups': 1704, 'old_label': None, 'order': 900, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 115], [3, 8], [5, 24], [6, 200], [10, 360], [15, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 4, 12, 4, 4], 'outer_gen_pows': [1, 20, 1, 0, 0], 'outer_gens': [[1, 2, 32, 780], [1, 42, 16, 780], [1, 3, 80, 312], [1, 2, 356, 80], [1, 302, 324, 580]], 'outer_group': '384.17956', 'outer_hash': 17956, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 20, 'outer_perms': [391972776938433487, 140894329609637281, 1778137698400697542, 1419396445699562280, 392035728778339080], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3):D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 4], [8, 10], [16, 8]], 'representations': {'PC': {'code': 5565290102957731138397953480015991, 'gens': [1, 2, 3, 5], 'pres': [6, -2, -2, -3, -5, -3, -5, 290, 404, 68, 1155, 7204, 12610, 118, 25925, 12971]}, 'Perm': {'d': 16, 'gens': [87657655681, 87657293544, 3, 3991683, 1482509952000, 6504]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^2:C_2^2', 'transitive_degree': 90, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [12, 12, 8, 60], 'aut_gens': [[1, 2, 4, 60], [733, 122, 656, 780], [361, 343, 880, 324], [541, 282, 328, 740], [757, 142, 44, 780]], 'aut_group': None, 'aut_hash': 2436627162132881625, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 345600, 'aut_permdeg': 90, 'aut_perms': [1317905993949449910595619579072711277353925821947402462258111618689098790614008122274708605306091730288780453792046726394595432097576765148, 1269567077681077699560137000948903663775642455131507690020963826412852985773859619641014997948047447771572393999517114689740433731455492806, 411907624817898892856726234153072531424231865265091162739970789103174503127107609169219617767980292552542683555021161260427483622621627787, 113116181324502536131239890590033757356086697392372493528661673651499081716327326725139190557286505778793158098015797064719646117818810003], 'aut_phi_ratio': 1440.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [2, 45, 2, 1], [3, 2, 4, 1], [5, 2, 4, 1], [5, 4, 4, 1], [6, 50, 4, 1], [10, 90, 4, 1], [15, 4, 16, 1], [15, 4, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3:S_3.C_{10}^2.C_{12}.C_4.C_2^2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': None, 'autcentquo_hash': 2436627162132881625, 'autcentquo_nilpotent': False, 'autcentquo_order': 345600, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3:S_3.C_{10}^2.C_{12}.C_4.C_2^2', 'cc_stats': [[1, 1, 1], [2, 25, 1], [2, 45, 2], [3, 2, 4], [5, 2, 4], [5, 4, 4], [6, 50, 4], [10, 90, 4], [15, 4, 48]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '900.131', 'commutator_count': 1, 'commutator_label': '225.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 131, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [2, 45, 1, 2], [3, 2, 1, 4], [5, 2, 2, 2], [5, 4, 2, 2], [6, 50, 1, 4], [10, 90, 2, 2], [15, 4, 2, 8], [15, 4, 4, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 756, 'exponent': 30, 'exponents_of_order': [2, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '900.131', 'hash': 131, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 2, 15, 15], 'inner_gens': [[1, 2, 16, 240], [1, 2, 44, 840], [49, 22, 4, 60], [721, 122, 4, 60]], 'inner_hash': 131, 'inner_nilpotent': False, 'inner_order': 900, 'inner_split': False, 'inner_tex': 'C_{15}^2:C_2^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 16], [4, 52]], 'label': '900.131', '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': 'C15^2:C2^2', 'ngens': 6, '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, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 72, 'number_divisions': 34, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 45, 'number_subgroup_classes': 144, 'number_subgroups': 1704, 'old_label': None, 'order': 900, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 115], [3, 8], [5, 24], [6, 200], [10, 360], [15, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 4, 12, 4, 4], 'outer_gen_pows': [1, 20, 1, 0, 0], 'outer_gens': [[1, 2, 32, 780], [1, 42, 16, 780], [1, 3, 80, 312], [1, 2, 356, 80], [1, 302, 324, 580]], 'outer_group': '384.17956', 'outer_hash': 17956, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 20, 'outer_perms': [391972776938433487, 140894329609637281, 1778137698400697542, 1419396445699562280, 392035728778339080], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3):D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 4], [8, 10], [16, 8]], 'representations': {'PC': {'code': 5565290102957731138397953480015991, 'gens': [1, 2, 3, 5], 'pres': [6, -2, -2, -3, -5, -3, -5, 290, 404, 68, 1155, 7204, 12610, 118, 25925, 12971]}, 'Perm': {'d': 16, 'gens': [87657655681, 87657293544, 3, 3991683, 1482509952000, 6504]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^2:C_2^2', 'transitive_degree': 90, 'wreath_data': None, 'wreath_product': False}
