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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1260.108', 'ambient_counter': 108, 'ambient_order': 1260, 'ambient_tex': 'C_{15}:D_{42}', 'central': False, 'central_factor': False, 'centralizer_order': 5, 'characteristic': False, 'core_order': 7, 'counter': 42, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1260.108.30.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '30.g1.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': 30, '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': '42.5', 'subgroup_hash': 5, 'subgroup_order': 42, 'subgroup_tex': 'D_{21}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1260.108', 'aut_centralizer_order': 4, 'aut_label': '30.g1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '252.26', 'aut_weyl_index': 24, 'centralizer': '252.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.g1.a1', '10.c1.a1'], 'contains': ['60.c1.a1', '90.c1.a1', '210.g1.a1'], 'core': '180.a1.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [1504, -1, 255, -1, 519, -1, 814, -1], 'generators': [3, 180, 844], 'label': '1260.108.30.g1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '10.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.g1.a1', 'old_label': '30.g1.a1', 'projective_image': '1260.108', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '30.g1.a1', 'subgroup_fusion': None, 'weyl_group': '42.5'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', '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, 21], 'aut_gens': [[1, 2], [1, 16], [1, 20], [33, 2]], 'aut_group': '252.26', 'aut_hash': 26, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 252, 'aut_permdeg': 10, 'aut_perms': [1, 203784, 2344323], 'aut_phi_ratio': 21.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 21, 1, 1], [3, 2, 1, 1], [7, 2, 3, 1], [21, 2, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times F_7', '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': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 21, 1], [3, 2, 1], [7, 2, 3], [21, 2, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '42.5', 'commutator_count': 1, 'commutator_label': '21.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '7.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 21, 1, 1], [3, 2, 1, 1], [7, 2, 3, 1], [21, 2, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 42, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[2, 1, 6]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '42.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 21], 'inner_gens': [[1, 40], [5, 2]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 42, 'inner_split': True, 'inner_tex': 'D_{21}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 10]], 'label': '42.5', 'linC_count': 6, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D21', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 12, 'number_divisions': 5, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 36, 'old_label': None, 'order': 42, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 21], [3, 2], [7, 6], [21, 12]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[1, 20]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 1], [12, 1]], 'representations': {'PC': {'code': 64747739, 'gens': [1, 2], 'pres': [3, -2, -3, -7, 241, 22, 326]}, 'GLFp': {'d': 2, 'p': 41, 'gens': [2756841, 289423]}, 'Perm': {'d': 10, 'gens': [41065, 3, 444984]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{21}', 'transitive_degree': 21, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [14, 12, 12, 6, 12], 'aut_gens': [[1, 2, 12], [9, 250, 852], [1, 310, 696], [1, 782, 1236], [1, 62, 732], [9, 242, 1056]], 'aut_group': None, 'aut_hash': 7474455982819576706, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6048, 'aut_permdeg': 28, 'aut_perms': [273003330896721486147622050438, 5486292860034512956929152692, 5559371469775811170423383896, 354028664934303312829488608, 278216273253665589171976544579], 'aut_phi_ratio': 21.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 21, 1, 1], [2, 63, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [5, 1, 4, 1], [6, 6, 1, 1], [6, 42, 1, 1], [7, 2, 3, 1], [10, 3, 4, 1], [10, 21, 4, 1], [10, 63, 4, 1], [14, 6, 3, 1], [15, 2, 4, 2], [15, 4, 4, 1], [21, 2, 6, 1], [21, 4, 3, 1], [21, 4, 6, 1], [30, 6, 4, 1], [30, 42, 4, 1], [35, 2, 12, 1], [42, 6, 6, 1], [70, 6, 12, 1], [105, 2, 24, 1], [105, 4, 12, 1], [105, 4, 24, 1], [210, 6, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times S_3^2\\times F_7', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '1512.792', 'autcentquo_hash': 792, 'autcentquo_nilpotent': False, 'autcentquo_order': 1512, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 21, 1], [2, 63, 1], [3, 2, 2], [3, 4, 1], [5, 1, 4], [6, 6, 1], [6, 42, 1], [7, 2, 3], [10, 3, 4], [10, 21, 4], [10, 63, 4], [14, 6, 3], [15, 2, 8], [15, 4, 4], [21, 2, 6], [21, 4, 9], [30, 6, 4], [30, 42, 4], [35, 2, 12], [42, 6, 6], [70, 6, 12], [105, 2, 24], [105, 4, 36], [210, 6, 24]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '252.36', 'commutator_count': 1, 'commutator_label': '63.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '5.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 108, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['42.5', 1], ['5.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 21, 1, 1], [2, 63, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [5, 1, 4, 1], [6, 6, 1, 1], [6, 42, 1, 1], [7, 2, 3, 1], [10, 3, 4, 1], [10, 21, 4, 1], [10, 63, 4, 1], [14, 6, 3, 1], [15, 2, 4, 2], [15, 4, 4, 1], [21, 2, 6, 1], [21, 4, 3, 1], [21, 4, 6, 1], [30, 6, 4, 1], [30, 42, 4, 1], [35, 2, 12, 1], [42, 6, 6, 1], [70, 6, 12, 1], [105, 2, 24, 1], [105, 4, 12, 1], [105, 4, 24, 1], [210, 6, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 210, 'exponents_of_order': [2, 2, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 5, 7], 'faithful_reps': [[4, 0, 24]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1260.108', 'hash': 108, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 6, 21], 'inner_gens': [[1, 10, 12], [5, 2, 492], [1, 782, 12]], 'inner_hash': 36, 'inner_nilpotent': False, 'inner_order': 252, 'inner_split': True, 'inner_tex': 'S_3\\times D_{21}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 96, 'irrQ_dim': 96, 'irrR_degree': 8, 'irrep_stats': [[1, 20], [2, 110], [4, 50]], 'label': '1260.108', 'linC_count': 600, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 40, 'linQ_dim': 14, 'linQ_dim_count': 40, 'linR_count': 336, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C15:D42', '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, 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': 30, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 180, 'number_divisions': 30, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 88, 'number_subgroup_classes': 88, 'number_subgroups': 792, 'old_label': None, 'order': 1260, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 87], [3, 8], [5, 4], [6, 48], [7, 6], [10, 348], [14, 18], [15, 32], [21, 48], [30, 192], [35, 24], [42, 36], [70, 72], [105, 192], [210, 144]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 2, 852], [1, 2, 624]], 'outer_group': '24.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [41064, 2403], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{12}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5], [6, 2], [8, 4], [12, 3], [16, 1], [24, 3], [48, 3], [96, 1]], 'representations': {'PC': {'code': 1233511898823537712845114356063123, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -3, -3, -5, -7, 121, 31, 146, 5913, 93, 3250, 178, 19451]}, 'GLZN': {'d': 2, 'p': 82, 'gens': [39389709, 22333806, 554893, 39293672, 23436503, 20400653]}, 'Perm': {'d': 18, 'gens': [21010447543680, 720, 33, 5760, 3991680, 397707843456000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}:D_{42}', 'transitive_degree': 210, 'wreath_data': None, 'wreath_product': False}
