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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '448.1028', 'ambient_counter': 1028, 'ambient_order': 448, 'ambient_tex': 'Q_8\\times D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 224, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '448.1028.2.b1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': ['C1'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '224.181', 'subgroup_hash': 181, 'subgroup_order': 224, 'subgroup_tex': 'Q_8\\times D_{14}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '448.1028', 'aut_centralizer_order': 4, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '16128.cf', 'aut_weyl_index': 8, 'centralizer': '112.a1', 'complements': ['224.d1'], 'conjugacy_class_count': 2, 'contained_in': ['1.a1'], 'contains': ['4.a1', '4.g1', '4.h1', '4.j1', '14.c1'], 'core': '2.b1', 'coset_action_label': None, 'count': 2, 'diagramx': [8842, 6771, 7871, 6416], 'generators': [1, 16, 56, 112, 2, 224], 'label': '448.1028.2.b1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.b1', 'normal_contained_in': ['1.a1'], 'normal_contains': ['4.a1', '4.g1', '4.h1'], 'normalizer': '1.a1', 'old_label': '2.b1', 'projective_image': '112.42', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.b1', 'subgroup_fusion': None, 'weyl_group': '112.42'}
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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': 3, 'aut_exponent': 84, 'aut_gen_orders': [6, 2, 2, 7, 2, 2, 6, 2, 4, 2], 'aut_gens': [[1, 2, 4, 16], [125, 2, 121, 208], [9, 2, 4, 208], [1, 114, 124, 208], [1, 34, 4, 16], [1, 2, 12, 208], [1, 2, 4, 216], [1, 2, 4, 48], [1, 10, 4, 208], [5, 10, 4, 208], [1, 114, 4, 208]], 'aut_group': '32256.bi', 'aut_hash': 5280945712516171064, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 32256, 'aut_permdeg': 19, 'aut_perms': [1466077363400264, 1466077363395409, 1466077363027282, 7180527132979200, 1466077363402396, 1466077450499778, 1782366430884480, 1466077323110400, 1466077363411780, 1466077363027200], 'aut_phi_ratio': 336.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 7, 4, 1], [4, 2, 6, 1], [4, 14, 6, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 2, 6, 1], [28, 4, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\wr D_6.F_7', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 7, 4], [4, 2, 6], [4, 14, 6], [7, 2, 3], [14, 2, 9], [28, 4, 18]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '56.12', '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', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 181, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['2.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 7, 1, 4], [4, 2, 1, 6], [4, 14, 1, 6], [7, 2, 3, 1], [14, 2, 3, 3], [28, 4, 3, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 23940, 'exponent': 28, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '112.42', 'hash': 181, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 2, 2, 7], 'inner_gens': [[1, 2, 12, 16], [1, 2, 4, 208], [9, 2, 4, 16], [1, 34, 4, 16]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': False, 'inner_tex': 'C_2\\times D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 28], [4, 6]], 'label': '224.181', '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': 'Q8*D14', 'ngens': 6, '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': 50, 'number_divisions': 30, 'number_normal_subgroups': 97, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 156, 'number_subgroups': 510, 'old_label': None, 'order': 224, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 31], [4, 96], [7, 6], [14, 18], [28, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 6, 2, 4, 6], 'outer_gen_pows': [1, 0, 0, 0, 0], 'outer_gens': [[1, 2, 125, 16], [5, 2, 113, 24], [121, 2, 4, 16], [1, 114, 4, 24], [1, 2, 116, 80]], 'outer_group': '576.8437', 'outer_hash': 8437, 'outer_nilpotent': False, 'outer_order': 576, 'outer_permdeg': 11, 'outer_perms': [41064, 403224, 11652480, 1704, 3669123], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,\\mathbb{Z}/4):C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4], [6, 8], [12, 2]], 'representations': {'PC': {'code': 138695379422028684100421, 'gens': [1, 2, 3, 5], 'pres': [6, -2, -2, -2, -2, -2, -7, 48, 218, 50, 3130, 88, 3467]}, 'GLZN': {'d': 2, 'p': 21, 'gens': [9325, 120406, 80557, 74096, 77323, 185354]}, 'Perm': {'d': 17, 'gens': [40447, 40320, 23906015481600, 45968909051520, 68436399225600, 973]}}, 'schur_multiplier': [2, 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': 'Q_8\\times D_{14}', 'transitive_degree': 112, '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': 3, 'aut_exponent': 84, 'aut_gen_orders': [14, 28, 4, 12, 12, 6], 'aut_gens': [[1, 2, 4, 112], [225, 94, 60, 168], [393, 270, 228, 336], [168, 270, 332, 57], [57, 274, 100, 337], [113, 86, 292, 392], [392, 86, 76, 337]], 'aut_group': None, 'aut_hash': 1868377163570225744, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 129024, 'aut_permdeg': 68, 'aut_perms': [1495984448124451469354323976600478487090591482973261170398222915629592050305517008231490147257427, 1350039588409080694987086251679591606108417169423492503640071335696504688892459497408312059613231, 1619768866547076400973717655033963473762228583966915721654024969313851628372691577372869698049788, 733731580232267025425923251405245949300134145909953889042739390616146902317541321742786555186896, 309911961175354975621560408931547604677937360400614699147121864883595295676150007717808822870029, 1554017081673501205775364053601729250669093015908383884234526383307131689155770921345305930051140], 'aut_phi_ratio': 672.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 14, 4, 1], [4, 2, 2, 1], [4, 2, 6, 1], [4, 4, 3, 1], [4, 28, 6, 1], [7, 2, 3, 1], [14, 2, 3, 3], [28, 2, 12, 1], [28, 4, 18, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.C_{42}.C_6.C_2^5', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '504.162', 'autcentquo_hash': 162, 'autcentquo_nilpotent': False, 'autcentquo_order': 504, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 14, 4], [4, 2, 8], [4, 4, 3], [4, 28, 6], [7, 2, 3], [14, 2, 9], [28, 2, 12], [28, 4, 36]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '112.42', 'commutator_count': 1, 'commutator_label': '28.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1028, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['56.5', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 14, 1, 4], [4, 2, 1, 8], [4, 4, 1, 3], [4, 28, 1, 6], [7, 2, 3, 1], [14, 2, 3, 3], [28, 2, 6, 2], [28, 4, 3, 6], [28, 4, 6, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 95760, 'exponent': 28, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '112.42', 'hash': 1028, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 2, 14, 2], 'inner_gens': [[1, 2, 4, 336], [1, 2, 108, 112], [1, 10, 4, 112], [225, 2, 4, 112]], 'inner_hash': 42, 'inner_nilpotent': False, 'inner_order': 112, 'inner_split': True, 'inner_tex': 'C_2^2\\times D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 56], [4, 13]], 'label': '448.1028', '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': 'Q8*D28', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 16, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 85, 'number_divisions': 40, 'number_normal_subgroups': 123, 'number_subgroup_autclasses': 76, 'number_subgroup_classes': 280, 'number_subgroups': 1284, 'old_label': None, 'order': 448, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 59], [4, 196], [7, 6], [14, 18], [28, 168]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 3, 2, 3, 2, 2], 'outer_gen_pows': [112, 0, 0, 28, 2, 0, 28, 0, 0], 'outer_gens': [[113, 282, 60, 112], [1, 226, 60, 112], [1, 226, 284, 112], [1, 30, 4, 112], [1, 2, 76, 112], [1, 226, 4, 112], [113, 58, 4, 225], [57, 2, 4, 168], [57, 2, 4, 112]], 'outer_group': '1152.157697', 'outer_hash': 2198351616017526314, 'outer_nilpotent': False, 'outer_order': 1152, 'outer_permdeg': 13, 'outer_perms': [1, 120, 479001600, 1037836800, 45360, 482630400, 3, 23, 7], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times D_4\\times S_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, 8], [4, 1], [6, 8], [12, 6], [24, 1]], 'representations': {'PC': {'code': 2729053653649000253475537695946804, 'gens': [1, 2, 3, 6], 'pres': [7, -2, -2, -2, -2, -7, -2, -2, 1568, 1143, 58, 1466, 80, 1691, 14117, 124]}, 'GLZN': {'d': 2, 'p': 84, 'gens': [28802010, 17188445, 38278904, 17488325, 38482886, 593713, 12004082]}, 'Perm': {'d': 19, 'gens': [357001369769809, 12460, 87109723, 119755729, 87091200, 5329, 7116370137676800]}}, 'schur_multiplier': [2, 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': 'Q_8\\times D_{28}', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
