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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1503.1', 'ambient_counter': 1, 'ambient_order': 1503, 'ambient_tex': 'C_{1503}', 'central': True, 'central_factor': False, 'centralizer_order': 1503, 'characteristic': True, 'core_order': 1, 'counter': 6, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '1503.1.1503.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '1503.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '1503.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1503, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{1503}', 'simple': False, 'solvable': True, 'special_labels': ['D', 'PC', 'U1', 'L1', 'D1', 'C3'], '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': '1503.1', 'aut_centralizer_order': None, 'aut_label': '1503.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': ['9.a1.a1', '501.a1.a1'], 'contains': [], 'core': '1503.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3836, 3836, 3836, 3836, 4554, 4554, 5379, 5379], 'generators': [], 'label': '1503.1.1503.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '1503.a1.a1', 'normal_contained_in': ['9.a1.a1', '501.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '1503.a1.a1', 'projective_image': '1503.1', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1503.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': True, 'abelian': True, 'abelian_quotient': '1503.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 498, 'aut_gen_orders': [2, 498], 'aut_gens': [[1], [836], [1069]], 'aut_group': '996.10', 'aut_hash': 10, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 996, 'aut_permdeg': 172, 'aut_perms': [7624971642806874085683452529554080177300705012568161351562792570466786218116214328567380623774817154661450829156514293423383121392803486041347542701635411625097574073437656743546196477232723853308434175342607370359330828633958382972591606630776365387831631582042848528920420939440, 92806600750703768408127743673691831830182471658218197119699754443132811082468349365355817387694918674225325135131932495777322005276327516569067127913493424541690091467462055699016854696387862012836428255692129527819681367932641926786665066911268852192861891283710168524336293973628616768942661142199183972830214], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1], [167, 1, 166, 1], [501, 1, 332, 1], [1503, 1, 996, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{498}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 498, 'autcent_group': '996.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 996, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{498}', '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], [3, 1, 2], [9, 1, 6], [167, 1, 166], [501, 1, 332], [1503, 1, 996]], 'center_label': '1503.1', 'center_order': 1503, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['3.1', '3.1', '167.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['167.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1], [167, 1, 166, 1], [501, 1, 332, 1], [1503, 1, 996, 1]], 'element_repr_type': 'PC', 'elementary': 501, 'eulerian_function': 1, 'exponent': 1503, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 83], 'factors_of_order': [3, 167], 'faithful_reps': [[1, 0, 996]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '501.1', 'hash': 1, 'hyperelementary': 501, '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': 996, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 1503]], 'label': '1503.1', 'linC_count': None, 'linC_degree': 1, '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': True, 'monomial': True, 'name': 'C1503', '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 1503, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 1503, 'order_factorization_type': 22, 'order_stats': [[1, 1], [3, 2], [9, 6], [167, 166], [501, 332], [1503, 996]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 498, 'outer_gen_orders': [2, 498], 'outer_gen_pows': [0, 0], 'outer_gens': [[836], [1069]], 'outer_group': '996.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 996, 'outer_permdeg': 172, 'outer_perms': [7624971642806874085683452529554080177300705012568161351562792570466786218116214328567380623774817154661450829156514293423383121392803486041347542701635411625097574073437656743546196477232723853308434175342607370359330828633958382972591606630776365387831631582042848528920420939440, 92806600750703768408127743673691831830182471658218197119699754443132811082468349365355817387694918674225325135131932495777322005276327516569067127913493424541690091467462055699016854696387862012836428255692129527819681367932641926786665066911268852192861891283710168524336293973628616768942661142199183972830214], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{498}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 176, 'pgroup': 0, 'primary_abelian_invariants': [9, 167], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 1], [2, 1], [6, 1], [166, 1], [332, 1], [996, 1]], 'representations': {'PC': {'code': 244464129015, 'gens': [1], 'pres': [3, -3, -3, -167, 9, 22]}, 'Perm': {'d': 176, 'gens': [9034370049068157396374400541504606602211925412766944560137692440274348533177677820572102011182207376112511725091041392687135521366701873173856119334851096455122952828880713643586461737681737316165317020290427763364810278708828088004082631737382484670143423524033068711563755520000000000000000000000000000000000000000000, 1494612823159220602834727446116490698483472390775123871527783668178402860636296923089193082650613568595129745165679275491241374525405329019506357391567179908243866583487687583237455390476159062872416219779074004057956107619164239470402730489650248203178564648960000000000000000000000000000000000000000, 2248899409081774192846131229535047586927404129299229326603186185499661438345538817758048639840589181448701734924339920801249679199330270840771805980744068331208927548322838972040445942225667085116338039104798844273739105555509368764019052217376473033259258274109161548985275514880000000000000000000000000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [1503], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{1503}', 'transitive_degree': 1503, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1503.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 498, 'aut_gen_orders': [2, 498], 'aut_gens': [[1], [836], [1069]], 'aut_group': '996.10', 'aut_hash': 10, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 996, 'aut_permdeg': 172, 'aut_perms': [7624971642806874085683452529554080177300705012568161351562792570466786218116214328567380623774817154661450829156514293423383121392803486041347542701635411625097574073437656743546196477232723853308434175342607370359330828633958382972591606630776365387831631582042848528920420939440, 92806600750703768408127743673691831830182471658218197119699754443132811082468349365355817387694918674225325135131932495777322005276327516569067127913493424541690091467462055699016854696387862012836428255692129527819681367932641926786665066911268852192861891283710168524336293973628616768942661142199183972830214], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1], [167, 1, 166, 1], [501, 1, 332, 1], [1503, 1, 996, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{498}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 498, 'autcent_group': '996.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 996, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{498}', '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], [3, 1, 2], [9, 1, 6], [167, 1, 166], [501, 1, 332], [1503, 1, 996]], 'center_label': '1503.1', 'center_order': 1503, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['3.1', '3.1', '167.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['167.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1], [167, 1, 166, 1], [501, 1, 332, 1], [1503, 1, 996, 1]], 'element_repr_type': 'PC', 'elementary': 501, 'eulerian_function': 1, 'exponent': 1503, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 83], 'factors_of_order': [3, 167], 'faithful_reps': [[1, 0, 996]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '501.1', 'hash': 1, 'hyperelementary': 501, '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': 996, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 1503]], 'label': '1503.1', 'linC_count': None, 'linC_degree': 1, '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': True, 'monomial': True, 'name': 'C1503', '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 1503, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 1503, 'order_factorization_type': 22, 'order_stats': [[1, 1], [3, 2], [9, 6], [167, 166], [501, 332], [1503, 996]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 498, 'outer_gen_orders': [2, 498], 'outer_gen_pows': [0, 0], 'outer_gens': [[836], [1069]], 'outer_group': '996.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 996, 'outer_permdeg': 172, 'outer_perms': [7624971642806874085683452529554080177300705012568161351562792570466786218116214328567380623774817154661450829156514293423383121392803486041347542701635411625097574073437656743546196477232723853308434175342607370359330828633958382972591606630776365387831631582042848528920420939440, 92806600750703768408127743673691831830182471658218197119699754443132811082468349365355817387694918674225325135131932495777322005276327516569067127913493424541690091467462055699016854696387862012836428255692129527819681367932641926786665066911268852192861891283710168524336293973628616768942661142199183972830214], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{498}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 176, 'pgroup': 0, 'primary_abelian_invariants': [9, 167], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 1], [2, 1], [6, 1], [166, 1], [332, 1], [996, 1]], 'representations': {'PC': {'code': 244464129015, 'gens': [1], 'pres': [3, -3, -3, -167, 9, 22]}, 'Perm': {'d': 176, 'gens': [9034370049068157396374400541504606602211925412766944560137692440274348533177677820572102011182207376112511725091041392687135521366701873173856119334851096455122952828880713643586461737681737316165317020290427763364810278708828088004082631737382484670143423524033068711563755520000000000000000000000000000000000000000000, 1494612823159220602834727446116490698483472390775123871527783668178402860636296923089193082650613568595129745165679275491241374525405329019506357391567179908243866583487687583237455390476159062872416219779074004057956107619164239470402730489650248203178564648960000000000000000000000000000000000000000, 2248899409081774192846131229535047586927404129299229326603186185499661438345538817758048639840589181448701734924339920801249679199330270840771805980744068331208927548322838972040445942225667085116338039104798844273739105555509368764019052217376473033259258274109161548985275514880000000000000000000000000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [1503], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{1503}', 'transitive_degree': 1503, 'wreath_data': None, 'wreath_product': False}
