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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '4032.fk', 'ambient_counter': 141, 'ambient_order': 4032, 'ambient_tex': 'C_{28}:(C_6\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 28, 'characteristic': False, 'core_order': 56, 'counter': 292, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4032.fk.36.u1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '36.u1.a1', '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': 36, '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': '112.38', 'subgroup_hash': 38, 'subgroup_order': 112, 'subgroup_tex': 'D_4\\times C_{14}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4032.fk', 'aut_centralizer_order': 56, 'aut_label': '36.u1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '48.52', 'aut_weyl_index': 336, 'centralizer': '144.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.a1.a1', '12.z1.a1', '18.h1.a1', '18.i1.a1', '18.j1.a1'], 'contains': ['72.a1.a1', '72.c1.a1', '72.i1.a1', '72.x1.a1', '72.be1.a1', '252.t1.a1'], 'core': '72.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [6441, -1, 4861, -1, 6963, -1, 4443, -1], 'generators': [1, 2016, 144, 504, 1008], 'label': '4032.fk.36.u1.a1', 'mobius_quo': None, 'mobius_sub': 2, 'normal_closure': '12.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '36.u1.a1', 'projective_image': '2016.dc', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36.u1.a1', 'subgroup_fusion': None, 'weyl_group': '48.52'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '56.13', '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, 2, 2, 3, 2, 2, 2], 'aut_gens': [[1, 2, 4], [1, 58, 53], [57, 31, 4], [1, 31, 60], [1, 58, 52], [1, 2, 100], [1, 59, 60], [1, 3, 4], [1, 58, 4]], 'aut_group': '384.17463', 'aut_hash': 17463, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 384, 'aut_permdeg': 13, 'aut_perms': [1614937704, 3437250504, 1480677144, 1611308184, 4, 2526048720, 15009120, 1611308160], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [4, 2, 2, 1], [7, 1, 6, 1], [14, 1, 6, 1], [14, 1, 12, 1], [14, 2, 24, 1], [28, 2, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2\\times C_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '192.1410', 'autcent_hash': 1410, 'autcent_nilpotent': True, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5:C_6', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [4, 2, 2], [7, 1, 6], [14, 1, 18], [14, 2, 24], [28, 2, 12]], 'center_label': '28.4', 'center_order': 28, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', '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': 38, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['7.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [4, 2, 1, 2], [7, 1, 6, 1], [14, 1, 6, 3], [14, 2, 6, 4], [28, 2, 6, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1197, 'exponent': 28, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '56.13', 'hash': 38, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2], 'inner_gens': [[1, 2, 4], [1, 2, 60], [1, 58, 4]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 56], [2, 14]], 'label': '112.38', 'linC_count': 384, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 8, 'linR_count': 24, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4*C14', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, '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': 10, 'number_conjugacy_classes': 70, 'number_divisions': 20, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 54, 'number_subgroups': 70, 'old_label': None, 'order': 112, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 11], [4, 4], [7, 6], [14, 66], [28, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[57, 2, 4], [1, 2, 52], [1, 2, 5], [1, 31, 37]], 'outer_group': '96.221', 'outer_hash': 221, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [1680, 744, 364560, 7298667], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 7], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [6, 8], [12, 2]], 'representations': {'PC': {'code': 15996467793029, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -7, 457, 42, 58]}, 'GLFq': {'d': 3, 'q': 8, 'gens': [51437379, 84073477, 33793538, 83940357, 67125252]}, 'Perm': {'d': 13, 'gens': [482630400, 83502720, 1037877120, 4320, 1037836800]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 14], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4\\times C_{14}', '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [42, 12, 6, 6, 12, 2], 'aut_gens': [[1, 2, 12, 1008, 2016], [2605, 3074, 3540, 3024, 1008], [1765, 1730, 1068, 1008, 3024], [757, 218, 2724, 1008, 3024], [2857, 3434, 3252, 3024, 1008], [1429, 170, 1308, 2016, 3024], [1, 794, 996, 1008, 3024]], 'aut_group': None, 'aut_hash': 2639570833013882725, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 52, 'aut_perms': [5090348801822909873985227045753012861240194960152751518173763008054, 4777318340158167222855005599620281194337791914556102146061828041088, 5201194580559512778902397913945996049672685577114125340358146050694, 43129165971178140795145620501011056587998888864922183686841464093533, 60136359571820568390858521990439775716551750190356635801587248203425, 43874044494537462858148113480083893678849516982157892561170747590166], 'aut_phi_ratio': 14.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 12, 2, 1], [2, 14, 2, 1], [2, 42, 2, 2], [3, 7, 1, 2], [3, 8, 1, 1], [3, 56, 1, 2], [4, 2, 1, 1], [4, 6, 1, 1], [4, 12, 2, 1], [4, 42, 2, 1], [4, 84, 1, 2], [6, 7, 1, 2], [6, 8, 1, 1], [6, 14, 2, 2], [6, 21, 1, 4], [6, 42, 2, 4], [6, 56, 1, 2], [6, 84, 2, 2], [6, 112, 2, 3], [7, 6, 1, 1], [12, 14, 1, 2], [12, 16, 1, 1], [12, 42, 1, 2], [12, 42, 2, 2], [12, 84, 1, 4], [12, 84, 2, 2], [12, 112, 1, 2], [14, 6, 1, 1], [14, 18, 1, 2], [14, 72, 2, 1], [21, 48, 1, 1], [28, 12, 1, 1], [28, 36, 1, 1], [28, 72, 2, 1], [42, 48, 1, 1], [84, 48, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_{14}\\times A_4).C_6.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '2016.dc', 'autcentquo_hash': 2868159204135431305, 'autcentquo_nilpotent': False, 'autcentquo_order': 2016, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_4\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [2, 12, 2], [2, 14, 2], [2, 42, 4], [3, 7, 2], [3, 8, 1], [3, 56, 2], [4, 2, 1], [4, 6, 1], [4, 12, 2], [4, 42, 2], [4, 84, 2], [6, 7, 2], [6, 8, 1], [6, 14, 4], [6, 21, 4], [6, 42, 8], [6, 56, 2], [6, 84, 4], [6, 112, 6], [7, 6, 1], [12, 14, 2], [12, 16, 1], [12, 42, 6], [12, 84, 8], [12, 112, 2], [14, 6, 1], [14, 18, 2], [14, 72, 2], [21, 48, 1], [28, 12, 1], [28, 36, 1], [28, 72, 2], [42, 48, 1], [84, 48, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '2016.dc', 'commutator_count': 1, 'commutator_label': '168.52', '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', '3.1', '7.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 141, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 12, 1, 2], [2, 14, 1, 2], [2, 42, 1, 4], [3, 7, 2, 1], [3, 8, 1, 1], [3, 56, 2, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 12, 1, 2], [4, 42, 1, 2], [4, 84, 1, 2], [6, 7, 2, 1], [6, 8, 1, 1], [6, 14, 2, 2], [6, 21, 2, 2], [6, 42, 2, 4], [6, 56, 2, 1], [6, 84, 2, 2], [6, 112, 1, 2], [6, 112, 2, 2], [7, 6, 1, 1], [12, 14, 2, 1], [12, 16, 1, 1], [12, 42, 2, 3], [12, 84, 2, 4], [12, 112, 2, 1], [14, 6, 1, 1], [14, 18, 1, 2], [14, 72, 1, 2], [21, 48, 1, 1], [28, 12, 1, 1], [28, 36, 1, 1], [28, 72, 1, 2], [42, 48, 1, 1], [84, 48, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1048320, 'exponent': 84, 'exponents_of_order': [6, 2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[36, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '2016.dc', 'hash': 3116638042093394956, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 6, 42, 2, 2], 'inner_gens': [[1, 2, 2868, 1008, 3024], [1, 2, 204, 1008, 3024], [1177, 818, 12, 2016, 3024], [1, 2, 3036, 1008, 2016], [1009, 1010, 1020, 1008, 2016]], 'inner_hash': 2868159204135431305, 'inner_nilpotent': False, 'inner_order': 2016, 'inner_split': True, 'inner_tex': 'C_2\\times S_4\\times F_7', 'inner_used': [1, 2, 3], 'irrC_degree': 36, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 36, 'irrep_stats': [[1, 24], [2, 18], [3, 24], [4, 3], [6, 10], [12, 5], [18, 4], [36, 1]], 'label': '4032.fk', 'linC_count': 576, 'linC_degree': 11, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 64, 'linQ_dim': 11, 'linQ_dim_count': 64, 'linR_count': 64, 'linR_degree': 11, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C28:(C6*S4)', 'ngens': 9, '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, 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': 65, 'number_characteristic_subgroups': 44, 'number_conjugacy_classes': 89, 'number_divisions': 63, 'number_normal_subgroups': 74, 'number_subgroup_autclasses': 584, 'number_subgroup_classes': 968, 'number_subgroups': 15908, 'old_label': None, 'order': 4032, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 227], [3, 134], [4, 284], [6, 1618], [7, 6], [12, 1192], [14, 186], [21, 48], [28, 192], [42, 48], [84, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [252, 0, 4], 'outer_gens': [[253, 2, 12, 1008, 2016], [1, 506, 12, 1008, 2016], [1, 2, 876, 1008, 2016]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [3, 8], [4, 7], [6, 14], [8, 1], [12, 5], [18, 4], [24, 1], [36, 1]], 'representations': {'PC': {'code': '7435739249139129686198194224644954209535354802301825751528597673556905445555571281640227099560088524', 'gens': [1, 2, 4, 8, 9], 'pres': [9, -2, -2, -3, -2, -2, -3, -7, -2, 2, 46, 103251, 3684, 4017, 102, 167404, 9193, 10012, 130, 129605, 22046, 10391, 212, 13623, 4560, 12130, 9115, 3076, 244952, 122489, 20447, 3446, 5156]}, 'Perm': {'d': 15, 'gens': [274468320311, 186818244480, 99632332800, 40447, 274467916800, 268252510843, 7660800, 3669120, 11652480]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{28}:(C_6\\times S_4)', 'transitive_degree': 84, 'wreath_data': None, 'wreath_product': False}
