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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '9604.m', 'ambient_counter': 13, 'ambient_order': 9604, 'ambient_tex': 'C_7\\wr C_4', 'central': False, 'central_factor': False, 'centralizer_order': 14, 'characteristic': False, 'core_order': 49, 'counter': 17, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '9604.m.49.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '49.a1', 'outer_equivalence': True, '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': 49, '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': '196.5', 'subgroup_hash': 5, 'subgroup_order': 196, 'subgroup_tex': 'C_7:C_{28}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '9604.m', 'aut_centralizer_order': None, 'aut_label': '49.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '686.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1'], 'contains': ['98.b1', '343.a1', '343.b1'], 'core': '196.a1', 'coset_action_label': None, 'count': 49, 'diagramx': [8905, -1, 9225, -1], 'generators': [7, 14, 5516, 4], 'label': '9604.m.49.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '49.a1', 'old_label': '49.a1', 'projective_image': '1372.31', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '49.a1', 'subgroup_fusion': None, 'weyl_group': '14.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '28.2', '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, 42], 'aut_gens': [[1, 28], [15, 28], [1, 84], [73, 28]], 'aut_group': '504.178', 'aut_hash': 178, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 504, 'aut_permdeg': 14, 'aut_perms': [744, 526943764, 19806758163], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 7, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 18, 1], [14, 1, 6, 1], [14, 2, 3, 1], [14, 2, 18, 1], [28, 7, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{14}:C_6^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_6', '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], [4, 7, 2], [7, 1, 6], [7, 2, 21], [14, 1, 6], [14, 2, 21], [28, 7, 12]], 'center_label': '14.2', 'center_order': 14, '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', '7.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['28.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 7, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 6, 3], [14, 1, 6, 1], [14, 2, 3, 1], [14, 2, 6, 3], [28, 7, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 28, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[2, 0, 18]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '98.3', 'hash': 5, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 168], [57, 28]], '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': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 28], [2, 42]], 'label': '196.5', 'linC_count': 18, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 5, 'linQ_dim': 12, 'linQ_dim_count': 3, 'linR_count': 27, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C7:C28', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 70, 'number_divisions': 14, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 18, 'number_subgroups': 36, 'old_label': None, 'order': 196, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 14], [7, 48], [14, 48], [28, 84]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[11, 28], [13, 84]], 'outer_group': '36.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 36, 'outer_permdeg': 10, 'outer_perms': [362883, 5184], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [4, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 4], [12, 7]], 'representations': {'PC': {'code': 128398020947, 'gens': [1, 4], 'pres': [4, -2, -2, -7, -7, 8, 21, 2691]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [390225, 536562, 957]}, 'Perm': {'d': 18, 'gens': [4032127, 378011776665600, 7983360, 973]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [28], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7:C_{28}', 'transitive_degree': 28, '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': '28.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 336, 'aut_gen_orders': [12, 24, 12, 48, 6], 'aut_gens': [[1, 28, 196, 1372], [843, 8204, 2324, 3276], [5225, 5992, 8260, 7560], [7391, 1932, 4816, 6076], [2217, 448, 8204, 8848], [6099, 3472, 1204, 8512]], 'aut_group': None, 'aut_hash': 1824229538155099556, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1185408, 'aut_permdeg': 692, 'aut_perms': 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'aut_phi_ratio': 288.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 49, 1, 1], [4, 343, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 18, 1], [7, 4, 12, 1], [7, 4, 72, 2], [7, 4, 432, 1], [14, 49, 6, 1], [14, 98, 3, 1], [14, 98, 18, 1], [28, 343, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_7^3.C_8.C_6^3.C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 336, 'autcentquo_group': None, 'autcentquo_hash': 5704297344339064290, 'autcentquo_nilpotent': False, 'autcentquo_order': 197568, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_7^3.C_8.C_6^2.C_2', 'cc_stats': [[1, 1, 1], [2, 49, 1], [4, 343, 2], [7, 1, 6], [7, 2, 21], [7, 4, 588], [14, 49, 6], [14, 98, 21], [28, 343, 12]], 'center_label': '7.1', 'center_order': 7, 'central_product': True, 'central_quotient': '1372.31', 'commutator_count': 1, 'commutator_label': '343.5', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '7.1', '7.1', '7.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1372.31', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 49, 1, 1], [4, 343, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 6, 3], [7, 4, 3, 4], [7, 4, 6, 96], [14, 49, 6, 1], [14, 98, 3, 1], [14, 98, 6, 3], [28, 343, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 28, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 0, 432]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '9604.m', 'hash': 6060614471244546186, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [4, 7, 7, 7], 'inner_gens': [[1, 1204, 2660, 8708], [197, 28, 196, 1372], [8709, 28, 196, 1372], [3837, 28, 196, 1372]], 'inner_hash': 31, 'inner_nilpotent': False, 'inner_order': 1372, 'inner_split': False, 'inner_tex': 'C_7^3:C_4', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': None, 'irrep_stats': [[1, 28], [2, 42], [4, 588]], 'label': '9604.m', 'linC_count': None, 'linC_degree': None, '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': False, 'monomial': True, 'name': 'C7wrC4', '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': 14, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 658, 'number_divisions': 114, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 54, 'number_subgroup_classes': 1004, 'number_subgroups': 5512, 'old_label': None, 'order': 9604, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 49], [4, 686], [7, 2400], [14, 2352], [28, 4116]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [6, 6, 24], 'outer_gen_pows': [0, 2, 2], 'outer_gens': [[13, 6972, 4480, 2800], [11, 812, 3920, 3976], [1, 2268, 7504, 9436]], 'outer_group': '864.3980', 'outer_hash': 3980, 'outer_nilpotent': True, 'outer_order': 864, 'outer_permdeg': 19, 'outer_perms': [3629040, 357001333113604, 34214091800359104], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{24}:C_6^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [4, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 4], [12, 11], [24, 96]], 'representations': {'PC': {'code': '18810850901272645466512595490751968577487', 'gens': [1, 4, 5, 6], 'pres': [6, -2, -2, -7, -7, 7, 7, 12, 31, 28899, 100137, 79804, 121810, 313493, 74603]}, 'Lie': [{'d': 1, 'q': 2401, 'family': 'ASigmaL'}], 'Perm': {'d': 28, 'gens': [91677670912242475163273354712, 111855893821342413774841997320]}}, 'schur_multiplier': [7], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [28], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_7\\wr C_4', 'transitive_degree': 28, 'wreath_data': ['C_7', 'C_4', '4T1'], 'wreath_product': True}
