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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1568.466', 'ambient_counter': 466, 'ambient_order': 1568, 'ambient_tex': 'C_{14}^2.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 56, 'characteristic': False, 'core_order': 28, 'counter': 25, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1568.466.28.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '28.d1', 'outer_equivalence': True, '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': 28, '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': '56.8', 'subgroup_hash': 8, 'subgroup_order': 56, 'subgroup_tex': 'C_2\\times C_{28}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1568.466', 'aut_centralizer_order': None, 'aut_label': '28.d1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '28.d1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['4.d1', '14.b1'], 'contains': ['56.a1', '56.g1', '196.b1'], 'core': '56.a1', 'coset_action_label': None, 'count': 28, 'diagramx': None, 'generators': [15, 16, 8, 784], 'label': '1568.466.28.d1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '14.b1', 'old_label': '28.d1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '28.d1', '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': '56.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 12], 'aut_gens': [[1, 2], [1, 26], [29, 2], [29, 47]], 'aut_group': '48.45', 'aut_hash': 45, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 48, 'aut_permdeg': 9, 'aut_perms': [24, 10824, 120964], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1], [7, 1, 6, 1], [14, 1, 6, 1], [14, 1, 12, 1], [28, 1, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '48.45', 'autcent_hash': 45, 'autcent_nilpotent': True, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6\\times D_4', '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], [4, 1, 4], [7, 1, 6], [14, 1, 18], [28, 1, 24]], 'center_label': '56.8', 'center_order': 56, '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', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [7, 1, 6, 1], [14, 1, 6, 3], [28, 1, 12, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 24, 'exponent': 28, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '28.4', 'hash': 8, '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, 56]], 'label': '56.8', 'linC_count': 576, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 24, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C28', '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 56, 'number_divisions': 12, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 56, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 4], [7, 6], [14, 18], [28, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 26], [29, 2], [29, 47]], 'outer_group': '48.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 9, 'outer_perms': [24, 10824, 120964], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [6, 4], [12, 2]], 'representations': {'PC': {'code': 41811557, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -7, 21, 34]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [243896, 682920]}, 'Perm': {'d': 13, 'gens': [11612160, 479001600, 4320, 3669120]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 28], '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_{28}', 'transitive_degree': 56, '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': 3, 'aut_exponent': 168, 'aut_gen_orders': [24, 12, 6, 24, 12, 14], 'aut_gens': [[1, 2, 16, 112], [1463, 738, 448, 800], [1123, 1166, 448, 840], [1355, 1422, 1344, 872], [687, 722, 672, 816], [837, 1290, 96, 344], [841, 386, 16, 1456]], 'aut_group': None, 'aut_hash': 1969245245644036849, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 451584, 'aut_permdeg': 112, 'aut_perms': [142948945965865554135660497110973846265627768631035453241506141856663590228245577846278661116475803662187284190315421553643532454729883777185304072142891943911745377417727594984783391, 20195868670861334120393959824130346752362525137554496455526897182928578903507276457434259002240663853443560733685906737425024231044463242541914191615933926899365845025615618666821125, 18899674834181245475424016832960858805465477215847985865208648964534210094162641952810598478944558503647737293206249302952785981527031783270585483658492855007018962682497470617524224, 58723277345372342989830979162273384589480896112010197556340748152929205319398743313197859735145663919097707303988829554132271071819687286605072764948498370533501847577879066999726334, 35550649708605632616504911241558944506218307126898056928442726368570071801647765783402101574845977654394034956895755767260998680180640692740280324985938086114195445889159045135272258, 67621051467061173005194030720482602197505683155437301238797657797437993348371510891734913232018564344349799786306099750207302915587134640404066762434552467716756713270636623471290736], 'aut_phi_ratio': 672.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 1, 2], [4, 28, 4, 1], [7, 2, 6, 1], [7, 4, 9, 1], [8, 98, 4, 1], [14, 2, 6, 1], [14, 2, 12, 1], [14, 4, 9, 1], [14, 4, 18, 1], [28, 4, 6, 2], [28, 4, 18, 2], [28, 28, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_7^2.(C_2^3\\times C_{12}).C_6.C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '14112.ba', 'autcentquo_hash': 3417587930551382414, 'autcentquo_nilpotent': False, 'autcentquo_order': 14112, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_7^2:D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 28, 4], [7, 2, 6], [7, 4, 9], [8, 98, 4], [14, 2, 18], [14, 4, 27], [28, 4, 48], [28, 28, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '392.20', 'commutator_count': 2, 'commutator_label': '196.7', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '7.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 466, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['784.57', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 28, 1, 4], [7, 2, 3, 2], [7, 4, 3, 3], [8, 98, 2, 2], [14, 2, 3, 6], [14, 4, 3, 9], [28, 4, 3, 4], [28, 4, 6, 6], [28, 28, 6, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2688, 'exponent': 56, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '392.41', 'hash': 466, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 4, 7, 7], 'inner_gens': [[1, 14, 96, 112], [5, 2, 96, 1456], [33, 34, 16, 112], [1, 226, 16, 112]], 'inner_hash': 20, 'inner_nilpotent': False, 'inner_order': 392, 'inner_split': False, 'inner_tex': 'D_{14}:D_7', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 54], [4, 84]], 'label': '1568.466', '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': 'C14^2.D4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 146, 'number_divisions': 46, 'number_normal_subgroups': 52, 'number_subgroup_autclasses': 62, 'number_subgroup_classes': 153, 'number_subgroups': 1070, 'old_label': None, 'order': 1568, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [4, 116], [7, 48], [8, 392], [14, 144], [28, 864]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [12, 6, 6, 6, 12], 'outer_gen_pows': [0, 0, 0, 672, 4], 'outer_gens': [[865, 1546, 64, 1464], [105, 106, 64, 112], [889, 322, 64, 336], [821, 1054, 64, 112], [3, 802, 448, 800]], 'outer_group': '1152.42995', 'outer_hash': 5990537374247802583, 'outer_nilpotent': False, 'outer_order': 1152, 'outer_permdeg': 96, 'outer_perms': [663521426070597943849493109567176827890017118920213183616650364728692467051630715948024803935045309207784532329839117983549279743846115187118269683634, 843871917167898833511239869663309061597431239759401214127162324181775996169244172307315145134576313213454240304063128866380826320483148916533560465850, 66375623268987221675712368586293288027371641409140782351408145610883958642213624442583605371681473703748131101028016180105541328507346110187351147353, 569253781220906541020825489095280183008213648404497895992019202233258078996848366059027926341657821027933994152871210886675698273647538382825868179846, 788943885170710943892238681891471727681775539927185254626281560941819044024333588513016265889965469706042846307091656477137159985916333068620030779821], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6.(C_3\\times D_4^2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 32, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 2], [6, 8], [12, 20], [24, 6]], 'representations': {'PC': {'code': 10771790018831922451734756063375888627382073, 'gens': [1, 2, 5, 6], 'pres': [7, -2, -2, -2, -2, -7, -2, -7, 56, 197, 36, 254, 58, 3364, 1691, 30588, 124, 32941]}, 'Perm': {'d': 32, 'gens': [9074144210848523173701637556601601, 17536036008583616295248854401029167, 1, 26057687342173993403825209970688000, 34556658493370206172007266574336000, 1482992582400, 50646]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_{14}^2.D_4', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
