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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.9757', 'ambient_counter': 9757, 'ambient_order': 1344, 'ambient_tex': 'C_{84}.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 56, 'counter': 157, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.9757.12.z1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.z1.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': 12, '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': '112.17', 'subgroup_hash': 17, 'subgroup_order': 112, 'subgroup_tex': 'C_7:Q_{16}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.9757', 'aut_centralizer_order': None, 'aut_label': '12.z1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '336.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.t1.b1', '6.g1.a1', '6.h1.a1', '6.k1.b1'], 'contains': ['24.f1.b1', '24.w1.a1', '24.y1.a1', '84.z1.b1'], 'core': '24.f1.b1', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [5, 8, 672, 338, 192], 'label': '1344.9757.12.z1.b1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.t1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '12.z1.b1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.z1.b1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [2, 2, 6, 28], 'aut_gens': [[1, 2, 16], [5, 6, 16], [1, 10, 16], [1, 2, 48], [13, 90, 16]], 'aut_group': '672.1093', 'aut_hash': 1093, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 672, 'aut_permdeg': 13, 'aut_perms': [6, 127, 200677806, 1132231368], 'aut_phi_ratio': 14.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 28, 1, 1], [7, 2, 3, 1], [8, 14, 2, 1], [14, 2, 3, 1], [28, 4, 3, 1], [28, 4, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4\\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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 4, 1], [4, 28, 1], [7, 2, 3], [8, 14, 2], [14, 2, 3], [28, 4, 9]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '56.7', 'commutator_count': 1, 'commutator_label': '28.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 17, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 28, 1, 1], [7, 2, 3, 1], [8, 14, 2, 1], [14, 2, 3, 1], [28, 4, 3, 1], [28, 4, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 56, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, -1, 3]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '28.3', 'hash': 17, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 4, 7], 'inner_gens': [[1, 14, 16], [5, 2, 96], [1, 34, 16]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': True, 'inner_tex': 'C_7:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 24, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 15], [4, 3]], 'label': '112.17', 'linC_count': 27, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 2, 'linQ_dim': 14, 'linQ_dim_count': 2, 'linR_count': 12, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C7:Q16', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_conjugacy_classes': 22, 'number_divisions': 10, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 18, 'number_subgroups': 52, 'old_label': None, 'order': 112, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [4, 34], [7, 6], [8, 28], [14, 6], [28, 36]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [4, 4], 'outer_gens': [[13, 2, 16], [13, 10, 48]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [24, 724], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1], [6, 2], [12, 2]], 'representations': {'PC': {'code': 11370390178593506981, 'gens': [1, 2, 5], 'pres': [5, -2, -2, -2, -2, -7, 40, 141, 26, 182, 42, 1209]}, 'GLZN': {'d': 2, 'p': 91, 'gens': [3316452, 40371124, 16927702, 29285190, 20346444]}, 'Perm': {'d': 23, 'gens': [1292878671028637817727, 2460805174576459025280, 3644911634129794062720, 4823231579504271763200, 973]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7:Q_{16}', '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': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 4, 12, 6, 6, 12, 2, 12], 'aut_gens': [[1, 2, 4, 16], [769, 2, 228, 1168], [1249, 682, 452, 656], [145, 346, 116, 368], [673, 682, 1132, 976], [1297, 338, 564, 944], [433, 1018, 1236, 1072], [97, 682, 460, 1328], [721, 338, 1236, 848]], 'aut_group': None, 'aut_hash': 9161810123478319507, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 32256, 'aut_permdeg': 48, 'aut_perms': [7398884183266007391493895269890145076901217921157999642939850, 6605922859111909520947083521092049640895330411267865262328463, 2124600924995266193374467288029688512925050484800225530693691, 6088795337555928168959951039929492715255969455919012266947269, 4756646506993872514111143120914100239738996186004800053031403, 6605956438249488774495738682827608585752995687680611715222429, 1090375159488928380541251133676535979850313516239301534113279, 107836441753519940908192103170173974855911913377861692720], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 1, 1], [2, 14, 2, 1], [2, 28, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 2, 1], [4, 12, 1, 1], [4, 14, 2, 1], [4, 28, 1, 1], [4, 84, 2, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 28, 2, 2], [7, 2, 3, 1], [8, 12, 2, 1], [8, 42, 2, 1], [8, 84, 1, 1], [12, 4, 1, 2], [12, 4, 4, 1], [12, 28, 2, 2], [14, 2, 3, 1], [14, 4, 3, 1], [14, 24, 3, 1], [21, 4, 3, 1], [28, 4, 3, 2], [28, 8, 6, 1], [28, 24, 3, 1], [42, 4, 3, 1], [42, 4, 6, 1], [56, 24, 6, 1], [84, 8, 3, 2], [84, 8, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', '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, 1], [2, 12, 1], [2, 14, 2], [2, 28, 1], [3, 2, 1], [4, 2, 2], [4, 4, 2], [4, 12, 1], [4, 14, 2], [4, 28, 1], [4, 84, 2], [6, 2, 1], [6, 4, 1], [6, 28, 4], [7, 2, 3], [8, 12, 2], [8, 42, 2], [8, 84, 1], [12, 4, 6], [12, 28, 4], [14, 2, 3], [14, 4, 3], [14, 24, 3], [21, 4, 3], [28, 4, 6], [28, 8, 6], [28, 24, 3], [42, 4, 9], [56, 24, 6], [84, 8, 18]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '672.1175', 'commutator_count': 1, 'commutator_label': '84.6', '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': 9757, '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, 1], [2, 12, 1, 1], [2, 14, 1, 2], [2, 28, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 2], [4, 12, 1, 1], [4, 14, 1, 2], [4, 28, 1, 1], [4, 84, 1, 2], [6, 2, 1, 1], [6, 4, 1, 1], [6, 28, 1, 2], [6, 28, 2, 1], [7, 2, 3, 1], [8, 12, 1, 2], [8, 42, 2, 1], [8, 84, 1, 1], [12, 4, 1, 2], [12, 4, 2, 2], [12, 28, 1, 2], [12, 28, 2, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 24, 3, 1], [21, 4, 3, 1], [28, 4, 3, 2], [28, 8, 3, 2], [28, 24, 3, 1], [42, 4, 3, 1], [42, 4, 6, 1], [56, 24, 3, 2], [84, 8, 3, 2], [84, 8, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 29877120, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, -1, 6]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '336.219', 'hash': 9757, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 4, 42], 'inner_gens': [[1, 2, 4, 880], [1, 2, 684, 688], [1, 10, 4, 1136], [481, 674, 228, 16]], 'inner_hash': 1175, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': True, 'inner_tex': 'D_{42}:C_2^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 96, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 44], [4, 32], [8, 10]], 'label': '1344.9757', '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.C2^4', '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': 41, 'number_characteristic_subgroups': 66, 'number_conjugacy_classes': 102, 'number_divisions': 52, 'number_normal_subgroups': 136, 'number_subgroup_autclasses': 352, 'number_subgroup_classes': 496, 'number_subgroups': 3508, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 71], [3, 2], [4, 248], [6, 118], [7, 6], [8, 192], [12, 136], [14, 90], [21, 12], [28, 144], [42, 36], [56, 144], [84, 144]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 2, 4, 464], [673, 2, 4, 16], [1, 2, 4, 688], [337, 338, 340, 304]], 'outer_group': '48.52', 'outer_hash': 52, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [40320, 3628800, 24, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 42, '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, 4], [6, 8], [8, 2], [12, 6], [24, 3], [48, 1]], 'representations': {'PC': {'code': 43867031880665960593204412217451238638643515288126577812402945547, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -7, 5385, 8218, 66, 10891, 2715, 35204, 13772, 11380, 116, 19973, 11157, 141, 46598, 7190, 222, 73735]}, 'Perm': {'d': 42, 'gens': [816438677858699393670790424925445379248139193607, 8701401292761284934381046272735131398, 24740528089383924376435516866838011, 34831114969901769894160164941278374, 43328436244259065405578978898675702, 50230513675554239111016559421853828, 303916116658416027343136804044800000000, 35085403596887862568986022075080631753113600000000]}}, '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': 'C_{84}.C_2^4', 'transitive_degree': 672, 'wreath_data': None, 'wreath_product': False}
