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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1344.922', 'ambient_counter': 922, 'ambient_order': 1344, 'ambient_tex': 'C_{28}.(C_6\\times D_4)', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 1, 'counter': 279, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1344.922.224.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '224.d1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 224, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '6.2', 'subgroup_hash': 2, 'subgroup_order': 6, 'subgroup_tex': 'C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.922', 'aut_centralizer_order': 384, 'aut_label': '224.d1', 'aut_quo_index': None, 'aut_stab_index': 28, 'aut_weyl_group': '1.1', 'aut_weyl_index': 10752, 'centralizer': '28.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['32.d1.a1', '112.b1.a1', '112.d1.a1', '112.k1.a1'], 'contains': ['448.a1.a1', '672.d1.a1'], 'core': '1344.a1.a1', 'coset_action_label': None, 'count': 28, 'diagramx': [8484, -1, 1379, -1, 1555, -1, 8490, -1], 'generators': [672, 16], 'label': '1344.922.224.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '28.b1.a1', 'old_label': '224.d1.a1', 'projective_image': '1344.922', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '224.d1.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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [5]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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, 1], [3, 1, 2], [6, 1, 2]], 'center_label': '6.2', 'center_order': 6, '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', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 6]], 'label': '6.2', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 6, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5]], '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': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2]], 'representations': {'PC': {'code': 21, 'gens': [1], 'pres': [2, -2, -3, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [73]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 56]}, 'Perm': {'d': 5, 'gens': [24, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6', 'transitive_degree': 6, '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 2, 6, 14, 6, 12, 6, 6], 'aut_gens': [[1, 2, 48, 96], [73, 62, 48, 912], [1201, 1034, 48, 912], [841, 194, 48, 1272], [805, 230, 48, 1056], [277, 470, 48, 144], [193, 422, 48, 528], [613, 1226, 48, 360], [769, 206, 48, 504], [793, 254, 48, 1128]], 'aut_group': None, 'aut_hash': 5229004673012084454, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10752, 'aut_permdeg': 68, 'aut_perms': [1252211669945184698914662494880803307641557648364387725860278601030299286677463673584303548130439, 666931705365590514899620651667001631088366884524556405431265110129948322820960746085318375535556, 784668842619280437345117575185162989348875990990036051198822799605801078300838698693426529281128, 2445197720164615810015241137424394983045431982894764274569197687426721163779912755681759371286813, 37023949469073462624891698257907294058270298171370213233346947926479498052196157128526996379453, 931088867006695969894603957697669317732884417483406982389281290539103457347092995556358376407797, 1625766431490878051970995190683029441696493987993586233746736672106933183536799416700798649967169, 1921890475458717582716174135987301933636008437370897923617233696260155183577890819633808349598132, 1447740897068207903793064257399715703223437698913633917696217665270533010590499367668071462490212], 'aut_phi_ratio': 28.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 28, 2, 1], [3, 7, 1, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 28, 2, 1], [4, 56, 1, 2], [6, 7, 1, 6], [6, 28, 1, 2], [6, 28, 2, 2], [7, 6, 1, 1], [8, 4, 2, 2], [12, 14, 1, 4], [12, 14, 2, 2], [12, 28, 2, 2], [12, 56, 1, 4], [14, 6, 1, 3], [14, 12, 2, 1], [24, 28, 2, 4], [28, 6, 2, 2], [28, 12, 2, 1], [56, 12, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_7.(C_2^5\\times C_6).C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 1], [2, 28, 2], [3, 7, 2], [4, 2, 4], [4, 28, 2], [4, 56, 2], [6, 7, 6], [6, 28, 6], [7, 6, 1], [8, 4, 4], [12, 14, 8], [12, 28, 4], [12, 56, 4], [14, 6, 3], [14, 12, 2], [24, 28, 8], [28, 6, 4], [28, 12, 2], [56, 12, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '336.123', 'commutator_count': 1, 'commutator_label': '56.8', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 922, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 28, 1, 2], [3, 7, 2, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 28, 1, 2], [4, 56, 1, 2], [6, 7, 2, 3], [6, 28, 2, 3], [7, 6, 1, 1], [8, 4, 2, 2], [12, 14, 2, 2], [12, 14, 4, 1], [12, 28, 2, 2], [12, 56, 2, 2], [14, 6, 1, 3], [14, 12, 2, 1], [24, 28, 4, 2], [28, 6, 2, 2], [28, 12, 2, 1], [56, 12, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 69888, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '168.47', 'hash': 922, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 12, 1, 14], 'inner_gens': [[1, 38, 48, 1272], [13, 2, 48, 1104], [1, 2, 48, 96], [217, 434, 48, 96]], 'inner_hash': 123, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': None, 'inner_tex': 'D_{28}:C_6', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 30], [4, 3], [6, 16], [12, 4]], 'label': '1344.922', 'linC_count': 96, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 16, 'linQ_dim': 16, 'linQ_dim_count': 48, 'linR_count': 192, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C28.(C6*D4)', 'ngens': 8, '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': 52, 'number_characteristic_subgroups': 64, 'number_conjugacy_classes': 77, 'number_divisions': 42, 'number_normal_subgroups': 72, 'number_subgroup_autclasses': 244, 'number_subgroup_classes': 304, 'number_subgroups': 1988, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 63], [3, 14], [4, 176], [6, 210], [7, 6], [8, 16], [12, 448], [14, 42], [24, 224], [28, 48], [56, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2], 'outer_gen_pows': [32, 416, 0, 592, 608], 'outer_gens': [[73, 62, 48, 912], [1201, 1034, 48, 912], [841, 194, 48, 1272], [805, 230, 48, 1056], [193, 422, 48, 528]], 'outer_group': '32.51', 'outer_hash': 51, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 32, 'outer_perms': [8286944364775336775771234688974663, 18571810710538201380391455935862960, 27179691438092070656247384601278720, 35679870403808096255989731269909896, 44178822125969133285376195933751740], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 7], [6, 4], [8, 2], [12, 4], [16, 1], [24, 1], [48, 1]], 'representations': {'PC': {'code': 4881353116382799530003615173439531714874570383191867705791647453, 'gens': [1, 2, 6, 7], 'pres': [8, 2, 2, 2, 2, 3, 2, 2, 7, 609, 41, 674, 66, 91, 71238, 30926, 12118, 7422, 3062, 166, 73735, 24591, 6167, 6175, 1575]}, 'Perm': {'d': 27, 'gens': [405964590934781901000343807, 886775988056277929791870080, 1307370579266313975644582400, 304, 7983360, 1690932141767855827749807360, 2127752503579100291781888000, 978]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{28}.(C_6\\times D_4)', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
