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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '3072.dk', 'ambient_counter': 89, 'ambient_order': 3072, 'ambient_tex': 'C_3\\times Q_{16}^2:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 6, 'counter': 690, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '3072.dk.256.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '256.f1.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': 256, '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': '12.2', 'subgroup_hash': 2, 'subgroup_order': 12, 'subgroup_tex': 'C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3072.dk', 'aut_centralizer_order': 256, 'aut_label': '256.f1', 'aut_quo_index': None, 'aut_stab_index': 16, 'aut_weyl_group': '4.2', 'aut_weyl_index': 4096, 'centralizer': '32.p1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['128.f1.a1', '128.v1.a1', '128.v1.b1'], 'contains': ['512.a1.a1', '768.f1.a1'], 'core': '512.a1.a1', 'coset_action_label': None, 'count': 16, 'diagramx': [5948, -1, 2681, -1, 6088, -1, 7086, -1], 'generators': [1833, 200, 192], 'label': '3072.dk.256.f1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '16.x1.a1', 'old_label': '256.f1.a1', 'projective_image': '512.60809', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '256.f1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '12.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2], 'aut_gens': [[1], [5], [7]], 'aut_group': '4.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [1, 6], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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], [3, 1, 2], [4, 1, 2], [6, 1, 2], [12, 1, 4]], 'center_label': '12.2', 'center_order': 12, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 12, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 12]], 'label': '12.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C12', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 12, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [12, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5], [7]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [4, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 3865, 'gens': [1], 'pres': [3, -2, -2, -3, 6, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20970031]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [619]}, 'Perm': {'d': 7, 'gens': [2400, 4, 744]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [12], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}', 'transitive_degree': 12, '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': 16, 'aut_gen_orders': [4, 4, 16, 4, 16, 16, 8], 'aut_gens': [[1, 2, 24, 384], [2741, 598, 1896, 528], [1057, 2402, 312, 1920], [2753, 2998, 552, 384], [1205, 850, 648, 2256], [2417, 1714, 24, 384], [1201, 2030, 1272, 1152], [721, 386, 120, 1920]], 'aut_group': None, 'aut_hash': 2163118224434657587, 'aut_nilpotency_class': 7, 'aut_nilpotent': True, 'aut_order': 16384, 'aut_permdeg': 66, 'aut_perms': [5650766066027760018757272012233847298369010343762559638248201925201842541199221560247694885, 272938906629259277941674274134844683017467284136933537564660658175698719417901637471781216397, 159437154891620704644291737987610484125836234192290379683899860801102273114878938250510390896, 258154072913455359752202339929685545134077179876928196485486428734527808740275471750372348133, 127841059916735936067200597317214315325116485494309551434495612965255756359165019744335808257, 518338849759323496049526436889570126467093362184338808115086035529320950281364422698036990498, 37183551964710303130004090226030691130002435799418975977825449884161589957511827039122352703], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 16, 2, 1], [2, 64, 1, 1], [3, 1, 2, 1], [4, 4, 1, 3], [4, 16, 2, 2], [4, 32, 1, 1], [4, 32, 2, 1], [6, 1, 2, 1], [6, 2, 2, 1], [6, 16, 4, 1], [6, 64, 2, 1], [8, 4, 2, 3], [8, 8, 1, 1], [8, 8, 2, 1], [8, 32, 2, 2], [8, 128, 2, 1], [12, 4, 2, 3], [12, 16, 4, 2], [12, 32, 2, 1], [12, 32, 4, 1], [16, 4, 4, 1], [16, 8, 2, 1], [16, 8, 4, 1], [16, 32, 4, 2], [24, 4, 4, 3], [24, 8, 2, 1], [24, 8, 4, 1], [24, 32, 4, 2], [24, 128, 4, 1], [48, 4, 8, 1], [48, 8, 4, 1], [48, 8, 8, 1], [48, 32, 8, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_4^2.C_2^4.C_2^4', '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': 16, 'autcentquo_group': '1024.dgh', 'autcentquo_hash': 3085455326441159288, 'autcentquo_nilpotent': True, 'autcentquo_order': 1024, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_8^2:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 16, 2], [2, 64, 1], [3, 1, 2], [4, 4, 3], [4, 16, 4], [4, 32, 3], [6, 1, 2], [6, 2, 2], [6, 16, 4], [6, 64, 2], [8, 4, 6], [8, 8, 3], [8, 32, 4], [8, 128, 2], [12, 4, 6], [12, 16, 8], [12, 32, 6], [16, 4, 4], [16, 8, 6], [16, 32, 8], [24, 4, 12], [24, 8, 6], [24, 32, 8], [24, 128, 4], [48, 4, 8], [48, 8, 12], [48, 32, 16]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '512.60809', 'commutator_count': 1, 'commutator_label': '128.445', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 89, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['1024.ddc', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 16, 1, 2], [2, 64, 1, 1], [3, 1, 2, 1], [4, 4, 1, 3], [4, 16, 1, 4], [4, 32, 1, 3], [6, 1, 2, 1], [6, 2, 2, 1], [6, 16, 2, 2], [6, 64, 2, 1], [8, 4, 2, 3], [8, 8, 1, 1], [8, 8, 2, 1], [8, 32, 1, 2], [8, 32, 2, 1], [8, 128, 1, 2], [12, 4, 2, 3], [12, 16, 2, 4], [12, 32, 2, 3], [16, 4, 4, 1], [16, 8, 2, 1], [16, 8, 4, 1], [16, 32, 2, 2], [16, 32, 4, 1], [24, 4, 4, 3], [24, 8, 2, 1], [24, 8, 4, 1], [24, 32, 2, 2], [24, 32, 4, 1], [24, 128, 2, 2], [48, 4, 8, 1], [48, 8, 4, 1], [48, 8, 8, 1], [48, 32, 4, 2], [48, 32, 8, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 559104, 'exponent': 48, 'exponents_of_order': [10, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 16], [8, 0, 12]], 'familial': False, 'frattini_label': '128.445', 'frattini_quotient': '24.15', 'hash': 132339587530496491, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 16, 'inner_gen_orders': [4, 4, 8, 8], 'inner_gens': [[1, 254, 552, 384], [61, 2, 456, 2256], [2929, 3026, 24, 384], [1, 1586, 24, 384]], 'inner_hash': 4561790705136276924, 'inner_nilpotent': True, 'inner_order': 512, 'inner_split': True, 'inner_tex': 'D_8\\wr C_2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 8, 'irrep_stats': [[1, 24], [2, 18], [4, 78], [8, 27]], 'label': '3072.dk', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 24, 'linQ_dim': 18, 'linQ_dim_count': 16, 'linR_count': 8, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3*Q16^2:C2^2', 'ngens': 11, 'nilpotency_class': 7, 'nilpotent': True, 'normal_counts': [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': 50, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 147, 'number_divisions': 64, 'number_normal_subgroups': 62, 'number_subgroup_autclasses': 496, 'number_subgroup_classes': 772, 'number_subgroups': 6266, 'old_label': None, 'order': 3072, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 99], [3, 2], [4, 172], [6, 198], [8, 432], [12, 344], [16, 320], [24, 864], [48, 640]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 768, 0], 'outer_gens': [[193, 2, 24, 384], [17, 202, 24, 384], [2257, 1814, 552, 384], [1729, 2, 120, 1920]], 'outer_group': '32.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [5040, 120, 362880, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 35, '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], [4, 12], [8, 13], [16, 11], [32, 5], [64, 1]], 'representations': {'PC': {'code': '37740966697794406388640067025389910533109846316249380388517117603714696246475412784826521080259241519609984', 'gens': [1, 2, 5, 9], 'pres': [11, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 16984, 5589, 56, 79334, 90, 1092, 30364, 12555, 2336, 158, 72869, 30112, 5571, 192, 140454, 70241, 5572, 226, 111691, 66558, 294, 58100, 63391, 328, 104565]}, 'GLFp': {'d': 4, 'p': 7, 'gens': [28487064012000, 703004153866, 14839374613412, 6253812280114, 16777895200295, 29905050279322, 17733832447600, 23908254970188, 21097394514521, 17262479569448, 9495688004000]}, 'Perm': {'d': 35, 'gens': [6297403454054624247140907240381246852744, 1488675443125762530510039685235031269908, 5964391209476500029361026096186108388564, 234721203489355397407478146156053680044, 3, 2989086149138654099001516473372459638347, 95788128003654390467264478912290853604, 1540775354640875070319152134023323799464, 1584191941518196058335743449730573468724, 2136186945083350046872051732586741235148, 4968341527342609473785738392577432323683]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times Q_{16}^2:C_2^2', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
