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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '400.103', 'ambient_counter': 103, 'ambient_order': 400, 'ambient_tex': 'C_5^2:D_8', 'central': False, 'central_factor': False, 'centralizer_order': 200, 'characteristic': False, 'core_order': 5, 'counter': 78, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '400.103.80.a1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '80.a1.c1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '80.15', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 15, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 80, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5:D_8', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '5.1', 'subgroup_hash': 1, 'subgroup_order': 5, 'subgroup_tex': 'C_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '400.103', 'aut_centralizer_order': 8000, 'aut_label': '80.a1', 'aut_quo_index': 1, 'aut_stab_index': 6, 'aut_weyl_group': '4.1', 'aut_weyl_index': 48000, 'centralizer': '2.a1.a1', 'complements': ['5.a1.f1', '5.a1.d1', '5.a1.b1', '5.a1.a1', '5.a1.e1'], 'conjugacy_class_count': 1, 'contained_in': ['16.a1.a1', '40.a1.c1', '40.b1.c1', '40.c1.c1'], 'contains': ['400.a1.a1'], 'core': '80.a1.c1', 'coset_action_label': None, 'count': 1, 'diagramx': [2089, 3842, 1315, 2257, 4287, 4792, 3069, 3508], 'generators': [84], 'label': '400.103.80.a1.c1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '80.a1.c1', 'normal_contained_in': ['16.a1.a1', '40.a1.c1'], 'normal_contains': ['400.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '80.a1.c1', 'projective_image': '400.103', 'quotient_action_image': '2.1', 'quotient_action_kernel': '40.10', 'quotient_action_kernel_order': 40, 'quotient_fusion': None, 'short_label': '80.a1.c1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '5.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [2]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', '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': 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], [5, 1, 4]], 'center_label': '5.1', 'center_order': 5, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['5.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '5.1', 'hash': 1, 'hyperelementary': 5, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 5]], 'label': '5.1', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 5, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 5, 'order_factorization_type': 1, 'order_stats': [[1, 1], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 5, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -5]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26329372]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131]}, 'Perm': {'d': 5, 'gens': [96]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [5], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5', 'transitive_degree': 5, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 120, 'aut_gen_orders': [20, 20, 12, 24, 24], 'aut_gens': [[1, 2, 20], [33, 194, 276], [313, 266, 112], [305, 90, 68], [213, 394, 388], [73, 390, 24]], 'aut_group': None, 'aut_hash': 111244745141461865, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192000, 'aut_permdeg': 104, 'aut_perms': [507643975499501423912500007842240132600995681722433157226687604706906979735077199594568453182203841951191577495384489783622391181598372078489812006323507493241176375, 3383478235631827175126868253174179568018227708490353119882420851749774762503153237929374472053544273670945466973579728080491750013169026598181620642420108449372784971, 9372169467301609522380282420383272951465835634858782664112521497994270598842368310349944033252131661089981702845057537230237433543796275389136986170052062070978390859, 3750045303205090944548941379070964257393562085051209142914664295106680842460624927431991290289686396390072677498767714168121658988596575776728901215430125278145056289, 6961226464751092847896354744207466988832535706279425134259896453196639100689973883561577762675656573365536147744826800396021427703290792585938428888669795938903425347], 'aut_phi_ratio': 1200.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [2, 100, 1, 1], [4, 2, 1, 1], [5, 2, 12, 1], [8, 50, 2, 1], [10, 2, 12, 1], [10, 4, 24, 1], [20, 4, 12, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times C_5:D_5).C_2^4.S_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '48000.bi', 'autcentquo_hash': 6908648364093524296, 'autcentquo_nilpotent': False, 'autcentquo_order': 48000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{10}^2:\\GL(2,5)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 1], [2, 100, 1], [4, 2, 1], [5, 2, 12], [8, 50, 2], [10, 2, 12], [10, 4, 24], [20, 4, 12]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '200.36', 'commutator_count': 1, 'commutator_label': '100.8', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 103, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [2, 100, 1, 1], [4, 2, 1, 1], [5, 2, 2, 6], [8, 50, 2, 1], [10, 2, 2, 6], [10, 4, 4, 6], [20, 4, 2, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 168, 'exponent': 40, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '100.15', 'hash': 103, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [2, 10, 10], 'inner_gens': [[1, 118, 380], [305, 2, 220], [41, 202, 20]], 'inner_hash': 36, 'inner_nilpotent': False, 'inner_order': 200, 'inner_split': True, 'inner_tex': 'C_{10}^2:C_2', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 51], [4, 12]], 'label': '400.103', 'linC_count': 2400, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 120, 'linQ_dim': 12, 'linQ_dim_count': 60, 'linR_count': 720, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5^2:D8', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_conjugacy_classes': 67, 'number_divisions': 30, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 33, 'number_subgroup_classes': 88, 'number_subgroups': 584, 'old_label': None, 'order': 400, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 105], [4, 2], [5, 24], [8, 100], [10, 120], [20, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [4, 8, 24, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 6, 260], [1, 250, 108], [1, 162, 112], [201, 102, 220], [201, 2, 20]], 'outer_group': '960.5693', 'outer_hash': 5693, 'outer_nilpotent': False, 'outer_order': 960, 'outer_permdeg': 26, 'outer_perms': [290570424132886124611068846, 49136032956504550302270840, 355144108339706615934727086, 15565162463473816018430767, 1], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times \\GL(2,5)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 13], [8, 12]], 'representations': {'PC': {'code': 19724429764189667215418517856451, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -5, -2, -2, -5, 1417, 31, 290, 9123, 2649, 69, 10804, 88, 11525]}, 'Perm': {'d': 18, 'gens': [21010055275756, 2813, 13307, 19342, 397620792172800, 1041828480]}}, 'schur_multiplier': [10], '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_5^2:D_8', 'transitive_degree': 200, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [2, 4, 4, 10], 'aut_gens': [[1, 2, 16], [1, 14, 16], [1, 2, 48], [13, 2, 16], [1, 74, 16]], 'aut_group': '320.1595', 'aut_hash': 1595, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 320, 'aut_permdeg': 11, 'aut_perms': [6, 1088767, 294, 10891567], 'aut_phi_ratio': 10.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [2, 20, 1, 1], [4, 2, 1, 1], [5, 2, 2, 1], [8, 10, 2, 1], [10, 2, 2, 1], [10, 4, 4, 1], [20, 4, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{10}.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '80.50', 'autcentquo_hash': 50, 'autcentquo_nilpotent': False, 'autcentquo_order': 80, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 1], [2, 20, 1], [4, 2, 1], [5, 2, 2], [8, 10, 2], [10, 2, 2], [10, 4, 4], [20, 4, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '40.8', 'commutator_count': 1, 'commutator_label': '20.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [2, 20, 1, 1], [4, 2, 1, 1], [5, 2, 2, 1], [8, 10, 2, 1], [10, 2, 2, 1], [10, 4, 4, 1], [20, 4, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 40, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 1, 2]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '20.4', 'hash': 15, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [2, 4, 5], 'inner_gens': [[1, 14, 16], [5, 2, 64], [1, 34, 16]], 'inner_hash': 8, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': True, 'inner_tex': 'C_5:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 11], [4, 2]], 'label': '80.15', 'linC_count': 18, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 3, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 10, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5:D8', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_conjugacy_classes': 17, 'number_divisions': 10, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 22, 'number_subgroup_classes': 22, 'number_subgroups': 74, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 25], [4, 2], [5, 4], [8, 20], [10, 20], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 10, 16], [1, 2, 32]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 3], [8, 2]], 'representations': {'PC': {'code': 2106151467538435, 'gens': [1, 2, 5], 'pres': [5, -2, -2, -2, -2, -5, 141, 26, 182, 42, 809]}, 'GLZN': {'d': 2, 'p': 35, 'gens': [1134191, 410744, 43121, 263649, 257256]}, 'Perm': {'d': 13, 'gens': [135001447, 498360960, 1086190560, 1611308160, 37]}}, 'schur_multiplier': [2], '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_5:D_8', 'transitive_degree': 40, 'wreath_data': None, 'wreath_product': False}
