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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1860.48', 'ambient_counter': 48, 'ambient_order': 1860, 'ambient_tex': 'C_{155}:A_4', 'central': True, 'central_factor': False, 'centralizer_order': 1860, 'characteristic': True, 'core_order': 1, 'counter': 20, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '1860.48.1860.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '1860.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '1860.48', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 48, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 1860, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'C_{155}:A_4', 'simple': False, 'solvable': True, 'special_labels': ['Phi', 'PC', 'U1', 'D2', 'C4'], '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': '1860.48', 'aut_centralizer_order': None, 'aut_label': '1860.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': ['60.a1.a1', '372.a1.a1', '620.a1.a1', '930.a1.a1'], 'contains': [], 'core': '1860.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8347, 2571, 2614, 6770, 6199, 5469, 3538, 4901], 'generators': [], 'label': '1860.48.1860.a1.a1', 'mobius_quo': 1, 'mobius_sub': 124, 'normal_closure': '1860.a1.a1', 'normal_contained_in': ['60.a1.a1', '372.a1.a1', '465.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '1860.a1.a1', 'projective_image': '1860.48', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1860.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': False, 'abelian_quotient': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1860, 'aut_gen_orders': [60, 60, 60], 'aut_gens': [[1, 3, 30], [1261, 939, 75], [1321, 936, 585], [451, 951, 1635]], 'aut_group': None, 'aut_hash': 6708314952976492426, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 44640, 'aut_permdeg': 128, 'aut_perms': [107537446930278894340550178807862835951950445494107763353324716765734450950221383134948528729641079092633660890484281987356157233842002744182245046555152763799863284850100937559241413352897330742081053097079026252313, 316731985207990086876241309864176328153585190714974756642105665705208338994109174680823427653425603545666843597432285577825523572821575483175382630933150198855948304820833779881550674494832310368842929850383884734804, 217301561833939135691554121726911872592500242113217348974707343798614457427791805764576262677792253140950282976577266722943057923103894993857466387511944683920341188716651754141575091585161143881951769213685734573999], 'aut_phi_ratio': 93.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 124, 1, 2], [5, 1, 4, 1], [10, 3, 4, 1], [15, 124, 4, 2], [31, 3, 10, 1], [62, 3, 30, 1], [155, 3, 40, 1], [310, 3, 120, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{31}.(C_{30}\\times A_4).C_2^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 930, 'autcentquo_group': None, 'autcentquo_hash': 2779991792977881782, 'autcentquo_nilpotent': False, 'autcentquo_order': 11160, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{31}:(C_{30}\\times A_4)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 124, 2], [5, 1, 4], [10, 3, 4], [15, 124, 8], [31, 3, 10], [62, 3, 30], [155, 3, 40], [310, 3, 120]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '372.11', 'commutator_count': 1, 'commutator_label': '124.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1', '31.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 48, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['372.11', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 124, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 124, 8, 1], [31, 3, 10, 1], [62, 3, 30, 1], [155, 3, 40, 1], [310, 3, 120, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 930, 'exponents_of_order': [2, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 31], 'factors_of_order': [2, 3, 5, 31], 'faithful_reps': [[3, 0, 120]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1860.48', 'hash': 48, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 186, 'inner_gen_orders': [3, 2, 62], 'inner_gens': [[1, 948, 165], [946, 3, 30], [1756, 3, 30]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 372, 'inner_split': True, 'inner_tex': 'C_{31}:A_4', 'inner_used': [1, 2, 3], 'irrC_degree': 3, 'irrQ_degree': 360, 'irrQ_dim': 360, 'irrR_degree': 6, 'irrep_stats': [[1, 15], [3, 205]], 'label': '1860.48', 'linC_count': 120, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 37, 'linQ_degree_count': 1, 'linQ_dim': 37, 'linQ_dim_count': 1, 'linR_count': 60, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C155:A4', '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 220, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 20, 'number_subgroups': 340, 'old_label': None, 'order': 1860, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 248], [5, 4], [10, 12], [15, 992], [31, 30], [62, 90], [155, 120], [310, 360]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [2, 60], 'outer_gen_pows': [0, 0], 'outer_gens': [[16, 3, 1830], [946, 939, 615]], 'outer_group': '120.31', 'outer_hash': 31, 'outer_nilpotent': True, 'outer_order': 120, 'outer_permdeg': 14, 'outer_perms': [6267300480, 127013793], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{60}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 40, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 1], [8, 1], [12, 1], [30, 1], [90, 1], [120, 1], [360, 1]], 'representations': {'PC': {'code': 20178318968781290264612729, 'gens': [1, 2, 4], 'pres': [5, 3, 2, 5, 2, 31, 9481, 26, 3303, 58, 7504]}, 'GLZN': {'d': 2, 'p': 124, 'gens': [2867625, 121070687, 184942625, 108208791, 1907121]}, 'Perm': {'d': 40, 'gens': [551667329564781559240213525681162415413066080, 33, 41040, 90720, 21486749411466781366110938107654143766552934400]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{155}:A_4', 'transitive_degree': 620, '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': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1860, 'aut_gen_orders': [60, 60, 60], 'aut_gens': [[1, 3, 30], [1261, 939, 75], [1321, 936, 585], [451, 951, 1635]], 'aut_group': None, 'aut_hash': 6708314952976492426, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 44640, 'aut_permdeg': 128, 'aut_perms': [107537446930278894340550178807862835951950445494107763353324716765734450950221383134948528729641079092633660890484281987356157233842002744182245046555152763799863284850100937559241413352897330742081053097079026252313, 316731985207990086876241309864176328153585190714974756642105665705208338994109174680823427653425603545666843597432285577825523572821575483175382630933150198855948304820833779881550674494832310368842929850383884734804, 217301561833939135691554121726911872592500242113217348974707343798614457427791805764576262677792253140950282976577266722943057923103894993857466387511944683920341188716651754141575091585161143881951769213685734573999], 'aut_phi_ratio': 93.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 124, 1, 2], [5, 1, 4, 1], [10, 3, 4, 1], [15, 124, 4, 2], [31, 3, 10, 1], [62, 3, 30, 1], [155, 3, 40, 1], [310, 3, 120, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{31}.(C_{30}\\times A_4).C_2^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 930, 'autcentquo_group': None, 'autcentquo_hash': 2779991792977881782, 'autcentquo_nilpotent': False, 'autcentquo_order': 11160, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{31}:(C_{30}\\times A_4)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 124, 2], [5, 1, 4], [10, 3, 4], [15, 124, 8], [31, 3, 10], [62, 3, 30], [155, 3, 40], [310, 3, 120]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '372.11', 'commutator_count': 1, 'commutator_label': '124.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1', '31.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 48, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['372.11', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 124, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 124, 8, 1], [31, 3, 10, 1], [62, 3, 30, 1], [155, 3, 40, 1], [310, 3, 120, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 930, 'exponents_of_order': [2, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 31], 'factors_of_order': [2, 3, 5, 31], 'faithful_reps': [[3, 0, 120]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1860.48', 'hash': 48, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 186, 'inner_gen_orders': [3, 2, 62], 'inner_gens': [[1, 948, 165], [946, 3, 30], [1756, 3, 30]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 372, 'inner_split': True, 'inner_tex': 'C_{31}:A_4', 'inner_used': [1, 2, 3], 'irrC_degree': 3, 'irrQ_degree': 360, 'irrQ_dim': 360, 'irrR_degree': 6, 'irrep_stats': [[1, 15], [3, 205]], 'label': '1860.48', 'linC_count': 120, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 37, 'linQ_degree_count': 1, 'linQ_dim': 37, 'linQ_dim_count': 1, 'linR_count': 60, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C155:A4', '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 220, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 20, 'number_subgroups': 340, 'old_label': None, 'order': 1860, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 248], [5, 4], [10, 12], [15, 992], [31, 30], [62, 90], [155, 120], [310, 360]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [2, 60], 'outer_gen_pows': [0, 0], 'outer_gens': [[16, 3, 1830], [946, 939, 615]], 'outer_group': '120.31', 'outer_hash': 31, 'outer_nilpotent': True, 'outer_order': 120, 'outer_permdeg': 14, 'outer_perms': [6267300480, 127013793], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{60}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 40, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 1], [8, 1], [12, 1], [30, 1], [90, 1], [120, 1], [360, 1]], 'representations': {'PC': {'code': 20178318968781290264612729, 'gens': [1, 2, 4], 'pres': [5, 3, 2, 5, 2, 31, 9481, 26, 3303, 58, 7504]}, 'GLZN': {'d': 2, 'p': 124, 'gens': [2867625, 121070687, 184942625, 108208791, 1907121]}, 'Perm': {'d': 40, 'gens': [551667329564781559240213525681162415413066080, 33, 41040, 90720, 21486749411466781366110938107654143766552934400]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{155}:A_4', 'transitive_degree': 620, 'wreath_data': None, 'wreath_product': False}
