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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '504.175', 'ambient_counter': 175, 'ambient_order': 504, 'ambient_tex': 'C_{21}:S_4', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 21, 'counter': 9, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '504.175.6.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.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': 6, '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': '84.4', 'subgroup_hash': 4, 'subgroup_order': 84, 'subgroup_tex': 'C_7:C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '504.175', 'aut_centralizer_order': 4, 'aut_label': '6.d1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '168.47', 'aut_weyl_index': 12, 'centralizer': '84.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.b1.a1'], 'contains': ['12.a1.a1', '18.c1.a1', '42.d1.a1'], 'core': '24.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [9736, -1, 9546, -1, 9258, -1, 9258, -1], 'generators': [43, 252, 6, 168], 'label': '504.175.6.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '6.d1.a1', 'projective_image': '168.46', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.d1.a1', 'subgroup_fusion': None, 'weyl_group': '28.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '12.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [2, 6, 14], 'aut_gens': [[1, 4], [3, 4], [1, 40], [61, 32]], 'aut_group': '168.47', 'aut_hash': 47, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 11, 'aut_perms': [7, 1829663, 8871256], 'aut_phi_ratio': 7.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 7, 2, 1], [6, 1, 2, 1], [7, 2, 3, 1], [12, 7, 4, 1], [14, 2, 3, 1], [21, 2, 6, 1], [42, 2, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times F_7', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 2], [4, 7, 2], [6, 1, 2], [7, 2, 3], [12, 7, 4], [14, 2, 3], [21, 2, 6], [42, 2, 6]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['28.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 7, 2, 1], [6, 1, 2, 1], [7, 2, 3, 1], [12, 7, 4, 1], [14, 2, 3, 1], [21, 2, 6, 1], [42, 2, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 84, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[2, 0, 6]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '42.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 52], [37, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 12, 'irrQ_dim': 24, 'irrR_degree': 4, 'irrep_stats': [[1, 12], [2, 18]], 'label': '84.4', 'linC_count': 6, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 10, 'linQ_dim_count': 3, 'linR_count': 9, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C7:C12', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_conjugacy_classes': 30, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 24, 'old_label': None, 'order': 84, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 14], [6, 2], [7, 6], [12, 28], [14, 6], [21, 12], [42, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 32], [3, 40]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [6, 2], [12, 2]], 'representations': {'PC': {'code': 1985247456347, 'gens': [1, 3], 'pres': [4, -2, -2, -3, -7, 8, 626, 46, 1155]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [10422, 24568]}, 'Perm': {'d': 14, 'gens': [482671584, 3, 1680, 7192321920]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [12], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7:C_{12}', 'transitive_degree': 84, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 6], 'aut_gens': [[1, 2, 42, 84], [253, 290, 42, 84], [23, 22, 252, 378], [39, 50, 252, 462]], 'aut_group': '2016.dc', 'aut_hash': 2868159204135431305, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2016, 'aut_permdeg': 44, 'aut_perms': [22713300979630411028047488106849792745526711941624486, 993510500026450821533345700440442113066972964960819736, 1620786337819167474437640960970499833077389446993908502], 'aut_phi_ratio': 14.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 42, 1, 1], [3, 1, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [4, 42, 1, 1], [6, 3, 2, 1], [6, 42, 2, 1], [7, 2, 3, 1], [12, 42, 2, 1], [14, 6, 3, 1], [21, 2, 6, 1], [21, 8, 6, 1], [21, 8, 12, 1], [42, 6, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_4\\times F_7', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '1008.886', 'autcentquo_hash': 886, 'autcentquo_nilpotent': False, 'autcentquo_order': 1008, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times F_7', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 42, 1], [3, 1, 2], [3, 8, 3], [4, 42, 1], [6, 3, 2], [6, 42, 2], [7, 2, 3], [12, 42, 2], [14, 6, 3], [21, 2, 6], [21, 8, 18], [42, 6, 6]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '168.46', 'commutator_count': 1, 'commutator_label': '84.10', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 175, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['168.46', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 42, 1, 1], [3, 1, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [4, 42, 1, 1], [6, 3, 2, 1], [6, 42, 2, 1], [7, 2, 3, 1], [12, 42, 2, 1], [14, 6, 3, 1], [21, 2, 6, 1], [21, 8, 6, 1], [21, 8, 12, 1], [42, 6, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 84, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[6, 0, 6]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '504.175', 'hash': 175, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 21, 2, 2], 'inner_gens': [[1, 40, 294, 84], [5, 2, 294, 378], [253, 254, 42, 84], [1, 296, 42, 84]], 'inner_hash': 46, 'inner_nilpotent': False, 'inner_order': 168, 'inner_split': True, 'inner_tex': 'C_7:S_4', 'inner_used': [1, 2, 3], 'irrC_degree': 6, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 12, 'irrep_stats': [[1, 6], [2, 30], [3, 6], [6, 9]], 'label': '504.175', 'linC_count': 144, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 4, 'linQ_dim': 11, 'linQ_dim_count': 4, 'linR_count': 54, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C21:S4', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 51, 'number_divisions': 16, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 48, 'number_subgroups': 380, 'old_label': None, 'order': 504, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 45], [3, 26], [4, 42], [6, 90], [7, 6], [12, 84], [14, 18], [21, 156], [42, 36]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 16, 294, 420], [1, 38, 42, 84]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [3, 2], [4, 1], [6, 3], [12, 2], [18, 1], [24, 1], [36, 1]], 'representations': {'PC': {'code': 8024830711826950003544445385, 'gens': [1, 2, 4, 5], 'pres': [6, -2, -3, -7, -2, 2, -3, 481, 43, 650, 7059, 3537, 5680, 88]}, 'GLZN': {'d': 2, 'p': 28, 'gens': [548825, 214622, 33321, 340271, 181306, 22065]}, 'Perm': {'d': 14, 'gens': [482670722, 840, 3, 7192321920, 7, 16]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{21}:S_4', 'transitive_degree': 84, 'wreath_data': None, 'wreath_product': False}
