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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '4300.c', 'ambient_counter': 3, 'ambient_order': 4300, 'ambient_tex': 'C_{10}\\times C_{430}', 'central': True, 'central_factor': False, 'centralizer_order': 4300, 'characteristic': False, 'core_order': 215, 'counter': 35, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '4300.c.20.a1.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '20.a1.f1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '20.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_{10}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '215.1', 'subgroup_hash': 1, 'subgroup_order': 215, 'subgroup_tex': 'C_{215}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4300.c', 'aut_centralizer_order': 120, 'aut_label': '20.a1', 'aut_quo_index': 1, 'aut_stab_index': 6, 'aut_weyl_group': '168.39', 'aut_weyl_index': 720, 'centralizer': '1.a1.a1', 'complements': ['215.a1.a1', '215.a1.c1', '215.a1.d1', '215.a1.e1', '215.a1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '10.a1.f1', '10.a1.l1', '10.a1.r1'], 'contains': ['100.a1.a1', '860.a1.f1'], 'core': '20.a1.f1', 'coset_action_label': None, 'count': 1, 'diagramx': [8608, 8608, 1084, 1084, 3808, 3808, 1585, 1585], 'generators': [2582, 100], 'label': '4300.c.20.a1.f1', 'mobius_quo': 1, 'mobius_sub': -2, 'normal_closure': '20.a1.f1', 'normal_contained_in': ['4.a1.a1', '10.a1.l1', '10.a1.f1', '10.a1.r1'], 'normal_contains': ['100.a1.a1', '860.a1.f1'], 'normalizer': '1.a1.a1', 'old_label': '20.a1.f1', 'projective_image': '20.5', 'quotient_action_image': '1.1', 'quotient_action_kernel': '20.5', 'quotient_action_kernel_order': 20, 'quotient_fusion': None, 'short_label': '20.a1.f1', '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': '215.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 84, 'aut_gen_orders': [2, 84], 'aut_gens': [[1], [171], [212]], 'aut_group': '168.39', 'aut_hash': 39, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 168, 'aut_permdeg': 16, 'aut_perms': [1321086412800, 6746343273], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [43, 1, 42, 1], [215, 1, 168, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{84}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.39', 'autcent_hash': 39, 'autcent_nilpotent': True, 'autcent_order': 168, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{84}', '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], [43, 1, 42], [215, 1, 168]], 'center_label': '215.1', 'center_order': 215, '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': ['5.1', '43.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['43.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [43, 1, 42, 1], [215, 1, 168, 1]], 'element_repr_type': 'PC', 'elementary': 215, 'eulerian_function': 1, 'exponent': 215, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [5, 43], 'faithful_reps': [[1, 0, 168]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '215.1', 'hash': 1, 'hyperelementary': 215, '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': 168, 'irrQ_dim': 168, 'irrR_degree': 2, 'irrep_stats': [[1, 215]], 'label': '215.1', 'linC_count': 168, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 46, 'linQ_degree_count': 1, 'linQ_dim': 46, 'linQ_dim_count': 1, 'linR_count': 84, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C215', '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': 215, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 215, 'order_factorization_type': 11, 'order_stats': [[1, 1], [5, 4], [43, 42], [215, 168]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [2, 84], 'outer_gen_pows': [0, 0], 'outer_gens': [[171], [212]], 'outer_group': '168.39', 'outer_hash': 39, 'outer_nilpotent': True, 'outer_order': 168, 'outer_permdeg': 16, 'outer_perms': [1321086412800, 6746343273], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{84}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 48, 'pgroup': 0, 'primary_abelian_invariants': [5, 43], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [42, 1], [168, 1]], 'representations': {'PC': {'code': 8987, 'gens': [1], 'pres': [2, -5, -43, 10]}, 'GLFp': {'d': 2, 'p': 431, 'gens': [10248063175]}, 'Perm': {'d': 48, 'gens': [1034492966044672722571857420614447919876790529556480000000000, 59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [215], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{215}', 'transitive_degree': 215, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4300.c', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 840, 'aut_gen_orders': [84, 168, 84, 42], 'aut_gens': [[1, 10], [2158, 213], [3016, 2453], [3879, 3356], [3017, 547]], 'aut_group': None, 'aut_hash': 4372067927312895334, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 120960, 'aut_permdeg': 69, 'aut_perms': [2516648415835065058348773931983744440720410637458936067671825994779904358353312734091673600000000, 2516532314666719300582401819818839375391154951632192523398540755306246756234188264435468585926400, 2480087463583371444584670102547379193138191070069983038867599486860280465488890703566626816000000, 4960105741404136443647088526266986312021303530846283800452072914877439833841983569893785600000000], 'aut_phi_ratio': 72.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 24, 1], [10, 1, 72, 1], [43, 1, 42, 1], [86, 1, 126, 1], [215, 1, 1008, 1], [430, 1, 3024, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{42}\\times S_3\\times \\GL(2,5)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 840, 'autcent_group': None, 'autcent_hash': 4372067927312895334, 'autcent_nilpotent': False, 'autcent_order': 120960, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_{42}\\times S_3\\times \\GL(2,5)', '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, 3], [5, 1, 24], [10, 1, 72], [43, 1, 42], [86, 1, 126], [215, 1, 1008], [430, 1, 3024]], 'center_label': '4300.c', 'center_order': 4300, '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': ['2.1', '2.1', '5.1', '5.1', '43.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['43.1', 1], ['5.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 6], [10, 1, 4, 18], [43, 1, 42, 1], [86, 1, 42, 3], [215, 1, 168, 6], [430, 1, 168, 18]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 44, 'exponent': 430, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 43], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '4300.c', 'hash': 1994011954018258, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 10], [1, 10]], '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': None, 'irrep_stats': [[1, 4300]], 'label': '4300.c', 'linC_count': None, 'linC_degree': 2, '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': 'C10*C430', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 4300, 'number_divisions': 56, 'number_normal_subgroups': 80, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 80, 'number_subgroups': 80, 'old_label': None, 'order': 4300, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [5, 24], [10, 72], [43, 42], [86, 126], [215, 1008], [430, 3024]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 840, 'outer_gen_orders': [420, 105, 84, 84], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[861, 51], [436, 1115], [1727, 1573], [2154, 3031]], 'outer_group': None, 'outer_hash': 4372067927312895334, 'outer_nilpotent': False, 'outer_order': 120960, 'outer_permdeg': 69, 'outer_perms': [4996592570289233071725955668595059678270675396820013195930106230180453231471247902819942400000000, 2516507448763696159898182402948253109073244613004406105509668586650945908046446371033283367475200, 4996651194223805687840701733244441319870164560556424994504296268724662961788293041009274104701400, 2516672391835584373506034395306577428932689968486839178828250172025694363278779657355657216000000], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_{42}\\times S_3\\times \\GL(2,5)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 57, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5, 5, 43], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 4], [4, 24], [42, 4], [168, 24]], 'representations': {'PC': {'code': '39988459697115487487', 'gens': [1, 3], 'pres': [5, -2, -5, -2, -5, -43, 10, 42, 118]}, 'GLFp': {'d': 2, 'p': 431, 'gens': [10168000210, 2081638081]}, 'Perm': {'d': 57, 'gens': [230843697339505628617695458715964099771805789389940151497296408420940313, 710998670045179751151054144953089682938954229455198813988050698240000000000]}}, 'schur_multiplier': [10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10, 430], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}\\times C_{430}', 'transitive_degree': 4300, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1, 2], [10, 13], [10, 15]], 'aut_group': '24.5', 'aut_hash': 5, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [127, 857], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 4, 1], [10, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.5', 'autcent_hash': 5, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times S_3', '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, 3], [5, 1, 4], [10, 1, 12]], 'center_label': '20.5', 'center_order': 20, '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', '2.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 1], [10, 1, 4, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '20.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 20]], 'label': '20.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 6, 'linQ_dim': 5, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C10', '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 20, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [5, 4], [10, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[10, 13], [10, 15]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [127, 857], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4]], 'representations': {'PC': {'code': 2179, 'gens': [1, 2], 'pres': [3, -2, -2, -5, 16]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706303489327]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [13320, 4001]}, 'Perm': {'d': 9, 'gens': [40320, 720, 96]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{10}', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
