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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '2592.lc', 'ambient_counter': 289, 'ambient_order': 2592, 'ambient_tex': 'C_3^2:D_6\\times S_4', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 1, 'counter': 1053, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2592.lc.648.l1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '648.l1.b1', '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': 648, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2592.lc', 'aut_centralizer_order': 48, 'aut_label': '648.l1', 'aut_quo_index': None, 'aut_stab_index': 108, 'aut_weyl_group': '1.1', 'aut_weyl_index': 5184, 'centralizer': '54.d1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['216.bl1.b1', '216.bm1.b1', '216.bn1.b1', '216.bx1.b1', '324.a1.b1', '324.x1.b1', '324.y1.a1'], 'contains': ['1296.b1.a1', '1296.c1.b1', '1296.h1.b1'], 'core': '2592.a1.a1', 'coset_action_label': None, 'count': 54, 'diagramx': [4132, -1, 5791, -1, 4542, -1, 6452, -1], 'generators': [1, 1901641], 'label': '2592.lc.648.l1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '6.a1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '54.d1.b1', 'old_label': '648.l1.b1', 'projective_image': '2592.lc', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '648.l1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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, 3]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': 4, 'aut_exponent': 12, 'aut_gen_orders': [4, 12, 6, 12], 'aut_gens': [[136242734, 571218611, 998808604], [3097302602, 1165733295, 1487849051], [1220810402, 771246735, 4050996611], [2414251581, 2884978083, 4090878128], [1053568085, 3171985212, 1487849055]], 'aut_group': None, 'aut_hash': 6886329914851957689, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5184, 'aut_permdeg': 24, 'aut_perms': [426639969234963880208774, 217011158808973489325216, 484237463953761295995622, 478004965469903885506701], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 1], [2, 9, 1, 1], [2, 9, 2, 1], [2, 27, 1, 1], [2, 27, 2, 1], [2, 54, 1, 1], [2, 54, 2, 1], [3, 2, 1, 1], [3, 6, 2, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 16, 1, 1], [3, 48, 2, 1], [3, 96, 1, 1], [4, 6, 1, 1], [4, 54, 1, 1], [4, 54, 2, 1], [6, 6, 1, 1], [6, 12, 1, 1], [6, 18, 1, 1], [6, 18, 2, 2], [6, 36, 1, 1], [6, 36, 2, 1], [6, 54, 1, 1], [6, 54, 2, 1], [6, 72, 1, 2], [6, 72, 2, 1], [6, 108, 1, 1], [6, 108, 2, 1], [6, 144, 1, 1], [6, 144, 2, 1], [12, 12, 1, 1], [12, 36, 2, 1], [12, 72, 1, 1], [12, 108, 1, 1], [12, 108, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '\\He_3.(D_4\\times S_4)', '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': 12, 'autcentquo_group': None, 'autcentquo_hash': 6886329914851957689, 'autcentquo_nilpotent': False, 'autcentquo_order': 5184, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\He_3.(D_4\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 6, 1], [2, 9, 3], [2, 27, 3], [2, 54, 3], [3, 2, 1], [3, 6, 2], [3, 8, 1], [3, 12, 1], [3, 16, 1], [3, 48, 2], [3, 96, 1], [4, 6, 1], [4, 54, 3], [6, 6, 1], [6, 12, 1], [6, 18, 5], [6, 36, 3], [6, 54, 3], [6, 72, 4], [6, 108, 3], [6, 144, 3], [12, 12, 1], [12, 36, 2], [12, 72, 1], [12, 108, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '2592.lc', 'commutator_count': 1, 'commutator_label': '324.130', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 289, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['108.17', 1], ['24.12', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 1], [2, 9, 1, 3], [2, 27, 1, 3], [2, 54, 1, 3], [3, 2, 1, 1], [3, 6, 1, 2], [3, 8, 1, 1], [3, 12, 1, 1], [3, 16, 1, 1], [3, 48, 1, 2], [3, 96, 1, 1], [4, 6, 1, 1], [4, 54, 1, 3], [6, 6, 1, 1], [6, 12, 1, 1], [6, 18, 1, 5], [6, 36, 1, 3], [6, 54, 1, 3], [6, 72, 1, 4], [6, 108, 1, 3], [6, 144, 1, 3], [12, 12, 1, 1], [12, 36, 1, 2], [12, 72, 1, 1], [12, 108, 1, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 725760, 'exponent': 12, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[18, 1, 4]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '864.4673', 'hash': 8934809208614015284, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 6, 6], 'inner_gens': [[136242734, 2167643644, 3173557091], [1978100762, 571218611, 4050996615], [3640180326, 1602293180, 998808604]], 'inner_hash': 8934809208614015284, 'inner_nilpotent': False, 'inner_order': 2592, 'inner_split': True, 'inner_tex': 'C_3^2:D_6\\times S_4', 'inner_used': [1, 2, 3], 'irrC_degree': 18, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 8], [2, 12], [3, 8], [4, 6], [6, 12], [8, 1], [12, 4], [18, 4]], 'label': '2592.lc', 'linC_count': 32, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 32, 'linQ_dim': 9, 'linQ_dim_count': 32, 'linR_count': 32, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:D6*S4', 'ngens': 3, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 40, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 55, 'number_divisions': 55, 'number_normal_subgroups': 52, 'number_subgroup_autclasses': 674, 'number_subgroup_classes': 1089, 'number_subgroups': 23670, 'old_label': None, 'order': 2592, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 279], [3, 242], [4, 168], [6, 1422], [12, 480]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[1931005581, 1576880768, 1643932083]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 6], [6, 12], [8, 1], [12, 4], [18, 4]], 'representations': {'PC': {'code': '1023796152765074764013614961840116095653161821019681265765750410703987197352222104818009766762775962408533', 'gens': [1, 2, 4, 6, 8], 'pres': [9, 2, 2, 3, 2, 3, 2, 3, 2, 3, 15984, 41221, 46, 6050, 52275, 29604, 9849, 102, 78844, 39973, 42773, 15566, 18491, 5540, 2147, 158, 72582, 36303, 10617, 4956, 155527, 23344, 35017, 12994, 214, 139976, 11699]}, 'Perm': {'d': 13, 'gens': [136242734, 571218611, 998808604]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:D_6\\times S_4', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
