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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '384.12567', 'ambient_counter': 12567, 'ambient_order': 384, 'ambient_tex': '(C_2\\times D_8).D_6', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 3, 'counter': 329, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '384.12567.32.u1.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '32.u1.e1', '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': 32, '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': '12.5', 'subgroup_hash': 5, 'subgroup_order': 12, 'subgroup_tex': 'C_2\\times C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.12567', 'aut_centralizer_order': None, 'aut_label': '32.u1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '8.p1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['16.o1.b1', '16.u1.a2', '16.x1.a1'], 'contains': ['64.d1.b1', '64.e1.a1', '64.e1.c1', '96.u1.e1'], 'core': '128.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': None, 'generators': [193, 2, 128], 'label': '384.12567.32.u1.e1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.x1.a1', 'old_label': '32.u1.e1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.u1.e1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [6, 9], [6, 11]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 31], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [6, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.4', 'autcent_hash': 4, 'autcent_nilpotent': False, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_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, 3], [3, 1, 2], [6, 1, 6]], 'center_label': '12.5', 'center_order': 12, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [6, 1, 2, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '12.5', 'hash': 5, '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, 12]], 'label': '12.5', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 6, 'linQ_dim': 3, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C6', '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 12, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 9], [6, 11]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4]], 'representations': {'PC': {'code': 273, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16507]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 2064]}, 'Perm': {'d': 7, 'gens': [720, 24, 4]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_6', 'transitive_degree': 12, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 2, 4, 12, 12, 4, 6, 2, 2], 'aut_gens': [[1, 2, 4, 16], [193, 11, 76, 377], [9, 10, 132, 368], [201, 3, 324, 369], [193, 299, 260, 113], [1, 290, 332, 216], [1, 98, 68, 88], [9, 107, 268, 121], [1, 2, 260, 376], [193, 2, 68, 272]], 'aut_group': None, 'aut_hash': 7341961092062084156, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 36, 'aut_perms': [322309957236172342460609751049522402008677, 116567807490906049257624635333986727616299, 323803752059369830602598997853695852829173, 292760381771594124602686277860921324863213, 200970245923384022838118112537671608735992, 200966307458243103168181773881988657986525, 53250108163244872366740778895588409393059, 115343443781724735330932532542922372339275, 165808608762968953913544170013755243030304], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 4, 4, 1], [2, 8, 1, 1], [3, 2, 1, 1], [4, 2, 2, 1], [4, 4, 1, 1], [4, 6, 2, 3], [4, 8, 1, 1], [4, 12, 1, 1], [4, 12, 4, 1], [4, 24, 1, 2], [6, 2, 1, 3], [6, 4, 2, 1], [6, 8, 4, 1], [6, 16, 1, 1], [8, 8, 2, 1], [8, 24, 2, 1], [12, 4, 2, 1], [12, 8, 1, 1], [12, 16, 1, 1], [24, 8, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^4.C_2^6.C_2^2)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '48.51', 'autcentquo_hash': 51, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 4], [2, 8, 1], [3, 2, 1], [4, 2, 2], [4, 4, 1], [4, 6, 6], [4, 8, 1], [4, 12, 5], [4, 24, 2], [6, 2, 3], [6, 4, 2], [6, 8, 4], [6, 16, 1], [8, 8, 2], [8, 24, 2], [12, 4, 2], [12, 8, 1], [12, 16, 1], [24, 8, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '96.209', 'commutator_count': 1, 'commutator_label': '24.9', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 12567, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 4], [2, 8, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 6, 2, 3], [4, 8, 1, 1], [4, 12, 1, 1], [4, 12, 2, 2], [4, 24, 1, 2], [6, 2, 1, 3], [6, 4, 1, 2], [6, 8, 1, 4], [6, 16, 1, 1], [8, 8, 1, 2], [8, 24, 1, 2], [12, 4, 1, 2], [12, 8, 1, 1], [12, 16, 1, 1], [24, 8, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 524160, 'exponent': 24, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '48.51', 'hash': 12567, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 2, 12], 'inner_gens': [[1, 2, 4, 24], [1, 2, 4, 112], [1, 2, 4, 80], [9, 290, 324, 16]], 'inner_hash': 209, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': None, 'inner_tex': 'D_4\\times D_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 20], [4, 14], [8, 1]], 'label': '384.12567', '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': '(C2*D8).D6', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 57, 'number_conjugacy_classes': 51, 'number_divisions': 44, 'number_normal_subgroups': 121, 'number_subgroup_autclasses': 282, 'number_subgroup_classes': 450, 'number_subgroups': 1390, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 31], [3, 2], [4, 160], [6, 62], [8, 64], [12, 32], [24, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 96, 96, 0], 'outer_gens': [[1, 2, 140, 272], [9, 298, 140, 184], [9, 10, 332, 80], [201, 106, 324, 272], [9, 106, 196, 88], [201, 107, 332, 377]], 'outer_group': '128.2163', 'outer_hash': 2163, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 32, 'outer_perms': [132107238905214132041533592652800636, 154865561314027884034792832827059605, 213324207635803878818110463815818404, 97577336683832521407715356068932696, 117352256485171404231771300184832069, 55062112444672077444606218263076166], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 12], [8, 4]], 'representations': {'PC': {'code': 616397814874840776531852132579683890177, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -2, 2, -2, -2, -3, 66, 964, 2252, 820, 116, 5389, 1941, 141, 4502, 166, 4119]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [2764825, 111361, 111745, 2157163, 1231525, 843445, 111631, 2821249]}, 'Perm': {'d': 19, 'gens': [6467845081426315, 1089305681600215, 1089305681590978, 14273539716017932, 20357887015065600, 20357887015090450, 13015, 3991680]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_8).D_6', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
