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gps_subgroup_search • Show schema
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{'Agroup': True, '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': 12, 'characteristic': False, 'core_order': 24, 'counter': 244, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4032.fk.28.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '28.b1.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': 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': '144.193', 'subgroup_hash': 193, 'subgroup_order': 144, 'subgroup_tex': 'C_2^2:C_6^2', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4032.fk', 'aut_centralizer_order': 24, 'aut_label': '28.b1', 'aut_quo_index': None, 'aut_stab_index': 14, 'aut_weyl_group': '48.48', 'aut_weyl_index': 336, 'centralizer': '336.e1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '14.a1.a1', '14.b1.b1', '14.d1.a1'], 'contains': ['56.a1.a1', '56.h1.a1', '84.v1.a1', '84.w1.a1', '84.x1.a1', '112.b1.a1'], 'core': '168.a1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [4466, -1, 1662, -1, 7413, -1, 1153, -1], 'generators': [7, 2016, 3024, 504, 4, 2352], 'label': '4032.fk.28.b1.a1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '28.b1.a1', 'projective_image': '2016.dc', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '28.b1.a1', 'subgroup_fusion': None, 'weyl_group': '48.48'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '36.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 3, 3, 3, 2, 2], 'aut_gens': [[1, 2, 12, 24], [1, 2, 12, 120], [1, 3, 12, 24], [1, 10, 72, 60], [1, 2, 84, 108], [7, 9, 12, 24], [1, 50, 12, 24], [1, 86, 12, 24], [1, 74, 12, 24]], 'aut_group': '864.4673', 'aut_hash': 4673, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 10, 'aut_perms': [379560, 374400, 5, 3, 1905840, 1941120, 23, 7], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 1, 1], [2, 3, 3, 1], [3, 1, 2, 1], [3, 4, 6, 1], [6, 1, 6, 1], [6, 3, 2, 1], [6, 3, 6, 1], [6, 4, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times S_3^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '36.10', 'autcent_hash': 10, 'autcent_nilpotent': False, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [3, 1, 2], [3, 4, 6], [6, 1, 6], [6, 3, 8], [6, 4, 18]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 193, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [3, 1, 2, 1], [3, 4, 2, 3], [6, 1, 2, 3], [6, 3, 2, 4], [6, 4, 2, 9]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '144.193', 'hash': 193, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 3, 2, 2], 'inner_gens': [[1, 2, 12, 24], [1, 2, 84, 108], [1, 74, 12, 24], [1, 86, 12, 24]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 36], [3, 12]], 'label': '144.193', 'linC_count': 144, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 18, 'linQ_dim': 5, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2:C6^2', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 48, 'number_divisions': 28, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 98, 'number_subgroups': 234, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 15], [3, 26], [6, 102]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 10, 72, 108], [7, 58, 72, 60], [7, 5, 72, 60]], 'outer_group': '72.46', 'outer_hash': 46, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 8, 'outer_perms': [744, 5761, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 16], [3, 4], [6, 4]], 'representations': {'PC': {'code': 25221789117966582921, 'gens': [1, 2, 4, 5], 'pres': [6, -2, -2, -3, -2, 2, -3, 31, 1017, 447, 1630, 286, 88]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [140827878942, 48233790121]}, 'GLFq': {'d': 3, 'q': 4, 'gens': [164358, 184618, 127769, 56040, 41268, 197379]}, 'Perm': {'d': 11, 'gens': [3669120, 7983360, 843, 840, 7, 16]}}, 'schur_multiplier': [2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 6], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2:C_6^2', 'transitive_degree': 36, '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}
