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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '12130560.c', 'ambient_counter': 3, 'ambient_order': 12130560, 'ambient_tex': '\\PGL(4,3)', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 1, 'counter': 202, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '12130560.c.12636.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12636.a1', 'outer_equivalence': True, 'perfect': True, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 12636, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': False, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '960.11358', 'subgroup_hash': 11358, 'subgroup_order': 960, 'subgroup_tex': 'C_2^4:A_5', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '12130560.c', 'aut_centralizer_order': None, 'aut_label': '12636.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '12130560.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['468.a1', '6318.a1'], 'contains': ['63180.J', '75816.B', '126360.T', '202176.A'], 'core': '12130560.a1', 'coset_action_label': None, 'count': 6318, 'diagramx': None, 'generators': [20487369, 42632213, 27293754, 30677096, 29506870, 29290330], 'label': '12130560.c.12636.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6318.a1', 'old_label': '12636.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12636.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 120, 'aut_gen_orders': [6, 6], 'aut_gens': [[388441, 121754], [930354, 1093683], [714800, 45416]], 'aut_group': '1920.240996', 'aut_hash': 3616852219197380499, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1920, 'aut_permdeg': 60, 'aut_perms': [5260633406035833409370463188202623211265120912979449752939078498516884323204788067, 3915276702189222760053434956999837618969495195557253827259354997052288502026395280], 'aut_phi_ratio': 7.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [2, 10, 1, 1], [2, 60, 1, 1], [3, 80, 1, 1], [4, 60, 1, 1], [4, 120, 1, 1], [5, 192, 2, 1], [6, 80, 1, 1], [6, 80, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:S_5', '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': '1920.240996', 'autcentquo_hash': 3616852219197380499, 'autcentquo_nilpotent': False, 'autcentquo_order': 1920, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^4:S_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [2, 10, 1], [2, 60, 1], [3, 80, 1], [4, 60, 1], [4, 120, 1], [5, 192, 2], [6, 80, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '960.11358', 'commutator_count': 2, 'commutator_label': '960.11358', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '60.5'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 11358, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [2, 10, 1, 1], [2, 60, 1, 1], [3, 80, 1, 1], [4, 60, 1, 1], [4, 120, 1, 1], [5, 192, 2, 1], [6, 80, 1, 1], [6, 80, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 285, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[5, 0, 2], [5, 1, 1], [10, 1, 2], [15, 1, 1], [20, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '960.11358', 'hash': 11358, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [4, 3], 'inner_gens': [[388441, 2227141], [1714342, 121754]], 'inner_hash': 11358, 'inner_nilpotent': False, 'inner_order': 960, 'inner_split': True, 'inner_tex': 'C_2^4:A_5', 'inner_used': [1, 2], 'irrC_degree': 5, 'irrQ_degree': 5, 'irrQ_dim': 5, 'irrR_degree': 5, 'irrep_stats': [[1, 1], [3, 2], [4, 1], [5, 4], [10, 2], [15, 1], [20, 1]], 'label': '960.11358', 'linC_count': 3, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 1, 'linQ_dim': 5, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C2^4:A5', 'ngens': 2, '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, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 12, 'number_divisions': 10, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 62, 'number_subgroup_classes': 63, 'number_subgroups': 1708, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 75], [3, 80], [4, 180], [5, 384], [6, 240]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [260844], 'outer_gens': [[1566018, 130134]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': None, 'perfect': True, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [5, 2], [6, 1], [10, 3], [15, 1], [20, 1]], 'representations': {'GLZ': {'b': 3, 'd': 5, 'gens': [516625397042, 428294947102]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [11107436, 30320728, 10024001, 32746561, 5128257, 6210625]}, 'Perm': {'d': 10, 'gens': [388441, 121754]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4:A_5', 'transitive_degree': 10, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4680, 'aut_gen_orders': [8, 8, 8], 'aut_gens': [[14408201, 9605495], [23793960, 32856903], [19251676, 29495479], [21546876, 29607407]], 'aut_group': None, 'aut_hash': 977983066756978958, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24261120, 'aut_permdeg': 1080, 'aut_perms': 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'aut_phi_ratio': 8.125, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1080, 1, 1], [2, 2106, 1, 1], [2, 5265, 1, 1], [3, 1040, 1, 1], [3, 6240, 1, 1], [3, 74880, 1, 1], [4, 4212, 1, 1], [4, 63180, 1, 1], [4, 189540, 1, 1], [4, 379080, 1, 1], [5, 303264, 1, 1], [6, 84240, 1, 1], [6, 112320, 1, 1], [6, 168480, 1, 1], [6, 336960, 1, 1], [6, 673920, 1, 1], [8, 63180, 2, 1], [8, 151632, 2, 1], [8, 379080, 1, 1], [8, 758160, 1, 1], [9, 449280, 1, 1], [10, 303264, 1, 1], [12, 336960, 1, 1], [12, 505440, 1, 1], [13, 466560, 2, 2], [20, 303264, 2, 1], [24, 505440, 2, 1], [26, 466560, 2, 2], [40, 303264, 2, 2]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(4,3).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': 4680, 'autcentquo_group': None, 'autcentquo_hash': 977983066756978958, 'autcentquo_nilpotent': False, 'autcentquo_order': 24261120, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PSL(4,3).C_2^2', 'cc_stats': [[1, 1, 1], [2, 1080, 1], [2, 2106, 1], [2, 5265, 1], [3, 1040, 1], [3, 6240, 1], [3, 74880, 1], [4, 4212, 1], [4, 63180, 1], [4, 189540, 1], [4, 379080, 1], [5, 303264, 1], [6, 84240, 1], [6, 112320, 1], [6, 168480, 1], [6, 336960, 1], [6, 673920, 1], [8, 63180, 2], [8, 151632, 2], [8, 379080, 1], [8, 758160, 1], [9, 449280, 1], [10, 303264, 1], [12, 336960, 1], [12, 505440, 1], [13, 466560, 4], [20, 303264, 2], [24, 505440, 2], [26, 466560, 4], [40, 303264, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '12130560.c', 'commutator_count': 1, 'commutator_label': '6065280.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '6065280.a'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1080, 1, 1], [2, 2106, 1, 1], [2, 5265, 1, 1], [3, 1040, 1, 1], [3, 6240, 1, 1], [3, 74880, 1, 1], [4, 4212, 1, 1], [4, 63180, 1, 1], [4, 189540, 1, 1], [4, 379080, 1, 1], [5, 303264, 1, 1], [6, 84240, 1, 1], [6, 112320, 1, 1], [6, 168480, 1, 1], [6, 336960, 1, 1], [6, 673920, 1, 1], [8, 63180, 2, 1], [8, 151632, 2, 1], [8, 379080, 1, 1], [8, 758160, 1, 1], [9, 449280, 1, 1], [10, 303264, 1, 1], [12, 336960, 1, 1], [12, 505440, 1, 1], [13, 466560, 4, 1], [20, 303264, 2, 1], [24, 505440, 2, 1], [26, 466560, 4, 1], [40, 303264, 4, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': None, 'exponent': 4680, 'exponents_of_order': [8, 6, 1, 1], 'factors_of_aut_order': [2, 3, 5, 13], 'factors_of_order': [2, 3, 5, 13], 'faithful_reps': [[39, 1, 2], [52, 0, 2], [52, 1, 1], [90, 1, 2], [130, 1, 1], [260, 1, 2], [351, 1, 2], [390, 1, 2], [416, 0, 6], [416, 1, 2], [468, 0, 2], [468, 1, 1], [520, 1, 1], [640, 0, 8], [729, 1, 2], [780, 1, 2], [1040, 0, 2], [1170, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12130560.c', 'hash': 8461112410546189005, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4680, 'inner_gen_orders': [2, 13], 'inner_gens': [[14408201, 9605441], [29819989, 9605495]], 'inner_hash': 8461112410546189005, 'inner_nilpotent': False, 'inner_order': 12130560, 'inner_split': True, 'inner_tex': '\\PGL(4,3)', 'inner_used': [1, 2], 'irrC_degree': 39, 'irrQ_degree': 39, 'irrQ_dim': 39, 'irrR_degree': 39, 'irrep_stats': [[1, 2], [39, 2], [52, 3], [90, 2], [130, 1], [260, 2], [351, 2], [390, 2], [416, 8], [468, 3], [520, 1], [640, 8], [729, 2], [780, 2], [1040, 2], [1170, 1]], 'label': '12130560.c', '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': False, 'metacyclic': False, 'monomial': False, 'name': 'PGL(4,3)', 'ngens': 2, '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, 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, 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, 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, 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, 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, 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, 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, 0, 0, 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'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 40, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [39, 2], [52, 1], [90, 2], [104, 1], [130, 1], [260, 2], [351, 2], [390, 2], [416, 2], [468, 1], [520, 1], [729, 2], [780, 2], [832, 1], [936, 1], [1170, 1], [1664, 1], [2080, 1], [2560, 2]], 'representations': {'Lie': [{'d': 4, 'q': 3, 'gens': [14408201, 9605495], 'family': 'PGL'}, {'d': 4, 'q': 3, 'family': 'PGammaL'}], 'Perm': {'d': 40, 'gens': [612318134292892751059980015582271582240530785125, 341668435346154550866383681368759885266699329565, 65971900822765570518592262312207230347168537455]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\PGL(4,3)', 'transitive_degree': 40, 'wreath_data': None, 'wreath_product': False}
