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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1344.1800', 'ambient_counter': 1800, 'ambient_order': 1344, 'ambient_tex': '(C_2\\times D_6):D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 42, 'counter': 102, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.1800.16.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '16.e1.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': '84.12', 'subgroup_hash': 12, 'subgroup_order': 84, 'subgroup_tex': 'C_3\\times D_{14}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.1800', 'aut_centralizer_order': 24, 'aut_label': '16.e1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '168.47', 'aut_weyl_index': 96, 'centralizer': '56.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.i1.a1', '8.l1.a1', '8.q1.a1'], 'contains': ['32.a1.a1', '32.f1.a1', '32.f1.b1', '48.e1.a1', '112.e1.a1'], 'core': '32.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [7602, -1, 7853, -1, 8107, -1, 7952, -1], 'generators': [3, 448, 192, 16], 'label': '1344.1800.16.e1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.b1.a1', 'old_label': '16.e1.a1', 'projective_image': '672.618', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.e1.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': '12.5', '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, 2], [1, 58], [1, 62], [19, 2]], '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], [2, 7, 2, 1], [3, 1, 2, 1], [6, 1, 2, 1], [6, 7, 4, 1], [7, 2, 3, 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], [2, 7, 2], [3, 1, 2], [6, 1, 2], [6, 7, 4], [7, 2, 3], [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': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 1, 2], [3, 1, 2, 1], [6, 1, 2, 1], [6, 7, 2, 2], [7, 2, 3, 1], [14, 2, 3, 1], [21, 2, 6, 1], [42, 2, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 42, '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': '1.1', 'frattini_quotient': '84.12', 'hash': 12, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 26], [61, 2]], '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, 12], [2, 18]], 'label': '84.12', 'linC_count': 6, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 6, 'linQ_dim': 8, 'linQ_dim_count': 6, 'linR_count': 21, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*D14', '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': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 20, 'number_subgroups': 56, 'old_label': None, 'order': 84, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 15], [3, 2], [6, 30], [7, 6], [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': [[43, 2], [1, 34]], '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': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [6, 2], [12, 2]], 'representations': {'PC': {'code': 166764352543211, 'gens': [1, 2], 'pres': [4, -2, -2, -3, -7, 209, 21, 626, 46, 1155]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [351, 2064, 1717]}, 'Perm': {'d': 12, 'gens': [3669840, 1, 30, 47255760]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times D_{14}', 'transitive_degree': 42, '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, 6, 6, 6, 14, 6, 12], 'aut_gens': [[1, 2, 8, 32], [385, 18, 456, 800], [1, 674, 8, 992], [773, 22, 1144, 352], [401, 6, 24, 1200], [593, 694, 472, 944], [977, 690, 8, 544], [21, 18, 1144, 1200]], 'aut_group': None, 'aut_hash': 4042710615256382448, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 48, 'aut_perms': [10447216415564792541917727151234615521929263811482272280255005, 8655918360467229349594069433461758662620974756637414732024584, 5314538610954872251704725276244747749396323751431960946685025, 8162707233160878003360484685844197387967172701637820683945635, 6630043538903111069612799666177460350642642933593465711379379, 10372723764415929875204699795084310626350698407362890411817815, 3113417356620550742873390375148024149263285533511444406485001], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 12, 1, 1], [2, 28, 1, 1], [2, 28, 2, 1], [2, 84, 1, 1], [3, 2, 1, 1], [4, 8, 1, 1], [4, 12, 1, 1], [4, 24, 1, 1], [4, 28, 1, 1], [4, 84, 1, 1], [4, 168, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 1, 2], [6, 28, 4, 1], [6, 56, 1, 1], [7, 2, 3, 1], [12, 8, 2, 1], [12, 56, 1, 1], [14, 2, 3, 1], [14, 4, 3, 3], [14, 24, 3, 1], [21, 4, 3, 1], [28, 8, 6, 1], [28, 24, 3, 1], [28, 24, 6, 1], [42, 4, 3, 1], [42, 4, 6, 1], [42, 8, 3, 2], [84, 8, 12, 1]], '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, 3], [2, 12, 1], [2, 28, 3], [2, 84, 1], [3, 2, 1], [4, 8, 1], [4, 12, 1], [4, 24, 1], [4, 28, 1], [4, 84, 1], [4, 168, 1], [6, 2, 3], [6, 4, 2], [6, 28, 4], [6, 56, 1], [7, 2, 3], [12, 8, 2], [12, 56, 1], [14, 2, 3], [14, 4, 9], [14, 24, 3], [21, 4, 3], [28, 8, 6], [28, 24, 9], [42, 4, 9], [42, 8, 6], [84, 8, 12]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '672.618', 'commutator_count': 1, 'commutator_label': '168.57', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 1800, '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, 3], [2, 12, 1, 1], [2, 28, 1, 3], [2, 84, 1, 1], [3, 2, 1, 1], [4, 8, 1, 1], [4, 12, 1, 1], [4, 24, 1, 1], [4, 28, 1, 1], [4, 84, 1, 1], [4, 168, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 1, 2], [6, 28, 2, 2], [6, 56, 1, 1], [7, 2, 3, 1], [12, 8, 2, 1], [12, 56, 1, 1], [14, 2, 3, 1], [14, 4, 3, 3], [14, 24, 3, 1], [21, 4, 3, 1], [28, 8, 6, 1], [28, 24, 3, 1], [28, 24, 6, 1], [42, 4, 3, 1], [42, 4, 6, 1], [42, 8, 3, 2], [84, 8, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 43008, 'exponent': 84, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, 0, 6]], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '168.50', 'hash': 1800, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 4, 2, 42], 'inner_gens': [[1, 6, 8, 432], [5, 2, 696, 48], [1, 690, 8, 928], [977, 18, 456, 32]], 'inner_hash': 618, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': True, 'inner_tex': 'D_6:D_{28}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 38], [4, 38], [8, 9]], 'label': '1344.1800', 'linC_count': 96, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 32, 'linQ_dim': 12, 'linQ_dim_count': 32, 'linR_count': 216, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*D6):D28', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 39, 'number_characteristic_subgroups': 58, 'number_conjugacy_classes': 93, 'number_divisions': 41, 'number_normal_subgroups': 58, 'number_subgroup_autclasses': 364, 'number_subgroup_classes': 396, 'number_subgroups': 4580, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 187], [3, 2], [4, 324], [6, 182], [7, 6], [12, 72], [14, 114], [21, 12], [28, 264], [42, 84], [84, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 2, 8, 928], [17, 2, 8, 32], [1, 2, 8, 992]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [720, 40320, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 6], [6, 4], [8, 2], [12, 6], [24, 4], [48, 1]], 'representations': {'PC': {'code': 18069265436422220186906418197864209942418011715302402571, 'gens': [1, 2, 4, 6], 'pres': [8, -2, -2, -2, -2, -2, 2, -3, -7, 97, 41, 11147, 211, 91, 20741, 1165, 5597, 141, 46598, 3614, 222, 73735]}, 'Perm': {'d': 18, 'gens': [21010447186129, 375221, 5593, 3036, 647, 18619, 3991680, 397707843456000]}}, 'schur_multiplier': [2, 2, 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_2\\times D_6):D_{28}', 'transitive_degree': 168, 'wreath_data': None, 'wreath_product': False}
