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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '672.1205', 'ambient_counter': 1205, 'ambient_order': 672, 'ambient_tex': 'C_{84}:C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': 336, 'characteristic': False, 'core_order': 28, 'counter': 35, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '672.1205.24.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '24.d1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '24.15', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 15, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 24, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2\\times C_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '28.2', 'subgroup_hash': 2, 'subgroup_order': 28, 'subgroup_tex': 'C_{28}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '672.1205', 'aut_centralizer_order': 5376, 'aut_label': '24.d1', 'aut_quo_index': 7, 'aut_stab_index': 4, 'aut_weyl_group': '12.5', 'aut_weyl_index': 21504, 'centralizer': '2.c1', 'complements': ['28.d1'], 'conjugacy_class_count': 4, 'contained_in': ['8.d1', '12.b1', '12.d1'], 'contains': ['48.b1', '168.c1'], 'core': '24.d1', 'coset_action_label': None, 'count': 4, 'diagramx': [319, 8816, 8494, 8129], 'generators': [506, 336, 96], 'label': '672.1205.24.d1', 'mobius_quo': 0, 'mobius_sub': 8, 'normal_closure': '24.d1', 'normal_contained_in': ['8.d1', '12.b1', '12.d1'], 'normal_contains': ['48.b1', '168.c1'], 'normalizer': '1.a1', 'old_label': '24.d1', 'projective_image': '336.225', 'quotient_action_image': '2.1', 'quotient_action_kernel': '12.5', 'quotient_action_kernel_order': 12, 'quotient_fusion': None, 'short_label': '24.d1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '28.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1], [13], [11]], 'aut_group': '12.5', 'aut_hash': 5, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 12, 'aut_permdeg': 7, 'aut_perms': [24, 723], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [7, 1, 6, 1], [14, 1, 6, 1], [28, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_6', '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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2], [7, 1, 6], [14, 1, 6], [28, 1, 12]], 'center_label': '28.2', 'center_order': 28, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '7.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [7, 1, 6, 1], [14, 1, 6, 1], [28, 1, 12, 1]], 'element_repr_type': 'PC', 'elementary': 14, 'eulerian_function': 1, 'exponent': 28, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7], 'faithful_reps': [[1, 0, 12]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '14.2', 'hash': 2, 'hyperelementary': 14, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 2, 'irrep_stats': [[1, 28]], 'label': '28.2', 'linC_count': 12, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 6, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C28', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 28, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 28, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [7, 6], [14, 6], [28, 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': [[13], [11]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [24, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [4, 7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 2], [12, 1]], 'representations': {'PC': {'code': 242573, 'gens': [1], 'pres': [3, -2, -2, -7, 6, 16]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [14163]}, 'Perm': {'d': 11, 'gens': [11612160, 4320, 3669120]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [28], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{28}', 'transitive_degree': 28, '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': '48.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 168, 'aut_gen_orders': [12, 12, 12, 12, 14], 'aut_gens': [[1, 2, 4, 8], [3, 2, 127, 43], [1, 338, 631, 251], [2, 3, 29, 523], [338, 339, 652, 522], [1, 2, 102, 569]], 'aut_group': None, 'aut_hash': 3465690225156616482, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 258048, 'aut_permdeg': 114, 'aut_perms': [1824599995864067448620808936183037985678744209559863744754832971295678469706177289636797887203163474463496098976946143666025983742826003817291766385436362521646878787035462580069732347221, 2408410068097108487022288806594077004474211415399424168493289187921022438883234439432727676466851020791611689069781173480199495499618030242688266434590336033256676237114086867037162797607, 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'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': None, 'autcent_hash': 1930889827755753443, 'autcent_nilpotent': False, 'autcent_order': 3072, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4.C_2^4.D_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '84.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 84, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 14, 8], [3, 1, 2], [4, 2, 4], [6, 1, 14], [6, 14, 16], [7, 2, 3], [12, 2, 8], [14, 2, 21], [21, 2, 6], [28, 2, 24], [42, 2, 42], [84, 2, 48]], 'center_label': '24.15', 'center_order': 24, 'central_product': True, 'central_quotient': '28.3', 'commutator_count': 1, 'commutator_label': '14.2', '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': 1205, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1], ['56.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 14, 1, 8], [3, 1, 2, 1], [4, 2, 1, 4], [6, 1, 2, 7], [6, 14, 2, 8], [7, 2, 3, 1], [12, 2, 2, 4], [14, 2, 3, 7], [21, 2, 6, 1], [28, 2, 6, 4], [42, 2, 6, 7], [84, 2, 12, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 239400, 'exponent': 84, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '336.225', 'hash': 1205, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [1, 1, 2, 14], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 440], [1, 2, 244, 8]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 28, 'inner_split': False, 'inner_tex': 'D_{14}', 'inner_used': [3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48], [2, 156]], 'label': '672.1205', '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': 'C84:C2^3', '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': 18, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 204, 'number_divisions': 64, 'number_normal_subgroups': 210, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 472, 'number_subgroups': 2204, 'old_label': None, 'order': 672, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 119], [3, 2], [4, 8], [6, 238], [7, 6], [12, 16], [14, 42], [21, 12], [28, 48], [42, 84], [84, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [6, 12, 12, 4], 'outer_gen_pows': [0, 0, 56, 0], 'outer_gens': [[337, 338, 7, 155], [1, 3, 172, 42], [337, 3, 172, 491], [339, 2, 6, 347]], 'outer_group': None, 'outer_hash': 1562395802877728387, 'outer_nilpotent': False, 'outer_order': 9216, 'outer_permdeg': 21, 'outer_perms': [12292976058494405784, 5385485579784978984, 5621284114150770121, 17306016362119518025], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^5.(C_6\\times S_4).C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 4], [6, 8], [12, 12], [24, 4]], 'representations': {'PC': {'code': 17320936972408056734229552116747, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, -2, -2, -2, -3, -7, 3097, 80, 1838, 102, 4387, 166, 7076]}, 'GLZN': {'d': 2, 'p': 28, 'gens': [548825, 285389, 22345, 603903, 329295, 32941, 22065]}, 'Perm': {'d': 18, 'gens': [21010447180927, 16, 5167, 11656, 3991680, 5160, 397707843456000]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{84}:C_2^3', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 84, 'aut_gen_orders': [2, 2, 4], 'aut_gens': [[1, 2, 4], [1, 3, 5], [1, 2, 20], [13, 14, 23]], 'aut_group': '336.209', 'aut_hash': 209, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 336, 'aut_permdeg': 9, 'aut_perms': [128184, 1, 225001], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [3, 1, 2, 1], [6, 1, 14, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times \\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '336.209', 'autcent_hash': 209, 'autcent_nilpotent': False, 'autcent_order': 336, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times \\GL(3,2)', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 1, 2], [6, 1, 14]], 'center_label': '24.15', 'center_order': 24, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 1], [6, 1, 2, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 13, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '24.15', 'hash': 15, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24]], 'label': '24.15', 'linC_count': 728, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 84, 'linQ_dim': 4, 'linQ_dim_count': 84, 'linR_count': 84, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C6', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 24, 'number_divisions': 16, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 32, 'number_subgroups': 32, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 14]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 3, 5], [1, 2, 20], [13, 14, 23]], 'outer_group': '336.209', 'outer_hash': 209, 'outer_nilpotent': False, 'outer_order': 336, 'outer_permdeg': 9, 'outer_perms': [128184, 1, 225001], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times \\GL(3,2)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8]], 'representations': {'PC': {'code': 8257, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -3, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931072, 7115648, 7233423]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [17150691, 34720862, 34603218, 15088919]}, 'Perm': {'d': 9, 'gens': [40320, 720, 24, 4]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_6', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
