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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '672.459', 'ambient_counter': 459, 'ambient_order': 672, 'ambient_tex': 'C_{12}.D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 21, 'counter': 56, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '672.459.16.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '16.c1.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': 16, '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': '672.459', 'aut_centralizer_order': 16, 'aut_label': '16.c1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '252.26', 'aut_weyl_index': 64, 'centralizer': '168.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.d1.a1'], 'contains': ['32.a1.a1', '48.c1.a1', '112.c1.a1'], 'core': '32.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [1037, -1, 85, -1, 526, -1, 530, -1], 'generators': [3, 96, 224], 'label': '672.459.16.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.h1.a1', 'old_label': '16.c1.a1', 'projective_image': '672.459', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.c1.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': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 12, 6, 2, 6, 14, 12], 'aut_gens': [[1, 2, 8], [409, 510, 248], [485, 114, 488], [577, 562, 40], [1, 338, 568], [77, 398, 152], [649, 622, 568], [605, 58, 488]], 'aut_group': None, 'aut_hash': 7631554720823566362, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 80, 'aut_perms': [5204784748174634808765850576100881499651724648831277429192355486043760556487398975038049055742211472484334788007656949, 46439891884191516001744130659743921437682127669953618461950956778489704125673595090019631673900950719406957380194600290, 20389775660776155193665733733917392303147863117544699765678374740824930151008417126614524619428634853946865658126395995, 736225865949515898225489224709352183701506118579123148813563667191887179818633891473544376884372106778449296007585145, 64552792078260085962808583788638586338211582971052880568326818509636484129117373744712430806249136999416044180160878658, 38720814857674036788040360388465157543421425549503255455017885886074396533205071767918023473735779686707969516227779198, 31274316811265478165497007173611206961267733474550140837554203481835458744212973920886847393128361489737595766862109800], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 84, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 28, 2, 1], [4, 84, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [7, 2, 3, 1], [8, 12, 2, 1], [12, 4, 1, 2], [12, 28, 4, 1], [14, 2, 3, 1], [14, 4, 3, 1], [21, 4, 3, 1], [28, 2, 6, 1], [28, 4, 3, 1], [42, 4, 3, 1], [42, 4, 6, 1], [56, 12, 12, 1], [84, 4, 6, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^4\\times C_6).C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': None, 'autcentquo_hash': 5451351369529179689, 'autcentquo_nilpotent': False, 'autcentquo_order': 2016, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3\\times S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 84, 1], [3, 2, 1], [4, 2, 2], [4, 28, 2], [4, 84, 1], [6, 2, 3], [7, 2, 3], [8, 12, 2], [12, 4, 2], [12, 28, 4], [14, 2, 3], [14, 4, 3], [21, 4, 3], [28, 2, 6], [28, 4, 3], [42, 4, 9], [56, 12, 12], [84, 4, 12]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '336.158', 'commutator_count': 1, 'commutator_label': '84.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 459, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 84, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 28, 1, 2], [4, 84, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [7, 2, 3, 1], [8, 12, 1, 2], [12, 4, 1, 2], [12, 28, 2, 2], [14, 2, 3, 1], [14, 4, 3, 1], [21, 4, 3, 1], [28, 2, 6, 1], [28, 4, 3, 1], [42, 4, 3, 1], [42, 4, 6, 1], [56, 12, 6, 2], [84, 4, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5376, 'exponent': 168, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[4, 0, 12]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '168.50', 'hash': 459, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 4, 42], 'inner_gens': [[1, 6, 440], [341, 2, 568], [241, 114, 8]], 'inner_hash': 158, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': True, 'inner_tex': 'C_6:D_{28}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 34], [4, 33]], 'label': '672.459', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 16, 'linQ_dim': 14, 'linQ_dim_count': 4, 'linR_count': 6, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C12.D28', 'ngens': 7, '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': 26, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 75, 'number_divisions': 30, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 100, 'number_subgroup_classes': 120, 'number_subgroups': 872, 'old_label': None, 'order': 672, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 87], [3, 2], [4, 144], [6, 6], [7, 6], [8, 24], [12, 120], [14, 18], [21, 12], [28, 24], [42, 36], [56, 144], [84, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 504], 'outer_gens': [[1, 2, 232], [1, 2, 344], [1, 6, 344], [169, 170, 296]], 'outer_group': '48.52', 'outer_hash': 52, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [3628800, 744, 24, 40323], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 3], [6, 4], [8, 1], [12, 6], [24, 1], [48, 1]], 'representations': {'PC': {'code': 1298900409321608295322556792163138994579298161405442699, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -2, -2, -2, -3, -7, 2352, 85, 36, 7142, 1780, 12323, 7962, 80, 7284, 8131, 102, 17477, 5388, 166, 28230]}, 'Perm': {'d': 26, 'gens': [621574835364635856419003, 358667549188915, 44141565574, 4410506897065, 5801702614492, 6758061133824000, 16754357281155028254720000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}.D_{28}', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
