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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '12450.a', 'ambient_counter': 1, 'ambient_order': 12450, 'ambient_tex': 'D_{25}\\times C_{249}', 'central': False, 'central_factor': True, 'centralizer_order': 249, 'characteristic': True, 'core_order': 4150, 'counter': 3, 'cyclic': False, 'direct': True, 'hall': 830, 'label': '12450.a.3.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '3.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '3.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 3, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '4150.b', 'subgroup_hash': 4, 'subgroup_order': 4150, 'subgroup_tex': 'D_{25}\\times C_{83}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '12450.a', 'aut_centralizer_order': None, 'aut_label': '3.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '50.a1.a1', 'complements': ['4150.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '15.a1.a1', '249.a1.a1'], 'core': '3.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7186, 1290, 1009, 1257, 2985, 4732, 2985, 1798], 'generators': [1, 150, 4980, 996], 'label': '12450.a.3.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '3.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['6.a1.a1', '249.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '3.a1.a1', 'projective_image': '150.2', 'quotient_action_image': '1.1', 'quotient_action_kernel': '3.1', 'quotient_action_kernel_order': 3, 'quotient_fusion': None, 'short_label': '3.a1.a1', 'subgroup_fusion': None, 'weyl_group': '50.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '166.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4100, 'aut_gen_orders': [164, 82, 410], 'aut_gens': [[1, 2], [3155, 186], [3985, 98], [2325, 3978]], 'aut_group': None, 'aut_hash': 12222580631130517, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 41000, 'aut_permdeg': 107, 'aut_perms': [6194090453340830805221210776193148317984010286602441386955378593271796531168490379263461227100178202781679155332107261892372385059393218469382616818841074538169814264205989, 1849560675628026658582271631060831192394317796090361994165349344463745533071545542925873336328811499554649765614889834050968016883086380767670123994965010876078563060031891, 1834287208443596099806602450439988911649124767593936912873752245299362941798916539098145022101629587459252681751830322782425696185518840359944791367507240538830381493058659], 'aut_phi_ratio': 25.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [5, 2, 2, 1], [25, 2, 10, 1], [83, 1, 82, 1], [166, 25, 82, 1], [415, 2, 164, 1], [2075, 2, 820, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{82}\\times D_{25}.C_{10}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 82, 'autcent_group': '82.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 82, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{82}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 100, 'autcentquo_group': '500.18', 'autcentquo_hash': 18, 'autcentquo_nilpotent': False, 'autcentquo_order': 500, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{25}:C_{20}', 'cc_stats': [[1, 1, 1], [2, 25, 1], [5, 2, 2], [25, 2, 10], [83, 1, 82], [166, 25, 82], [415, 2, 164], [2075, 2, 820]], 'center_label': '83.1', 'center_order': 83, 'central_product': True, 'central_quotient': '50.1', 'commutator_count': 1, 'commutator_label': '25.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '5.1', '5.1', '83.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['50.1', 1], ['83.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [5, 2, 2, 1], [25, 2, 10, 1], [83, 1, 82, 1], [166, 25, 82, 1], [415, 2, 164, 1], [2075, 2, 820, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 252, 'exponent': 4150, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 5, 41], 'factors_of_order': [2, 5, 83], 'faithful_reps': [[2, 0, 820]], 'familial': False, 'frattini_label': '5.1', 'frattini_quotient': '830.1', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 50, 'inner_gen_orders': [2, 25], 'inner_gens': [[1, 998], [3155, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 50, 'inner_split': False, 'inner_tex': 'D_{25}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 1640, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 166], [2, 996]], 'label': '4150.b', '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': True, 'monomial': True, 'name': 'D25*C83', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 1162, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 68, 'old_label': None, 'order': 4150, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 25], [5, 4], [25, 20], [83, 82], [166, 2050], [415, 328], [2075, 1640]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 410, 'outer_gen_orders': [2, 410], 'outer_gen_pows': [1, 0], 'outer_gens': [[1, 1164], [1, 1372]], 'outer_group': '820.20', 'outer_hash': 20, 'outer_nilpotent': True, 'outer_order': 820, 'outer_permdeg': 50, 'outer_perms': [258623241511168180642964355153611979969197632389120000000000, 608540612336810912965817692809489337748525225254846464000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{410}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 108, 'pgroup': 0, 'primary_abelian_invariants': [2, 83], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 2], [4, 1], [20, 1], [82, 2], [328, 1], [1640, 1]], 'representations': {'PC': {'code': '439081705905739340901875', 'gens': [1, 2], 'pres': [4, -2, -5, -5, -83, 7985, 45, 10082, 70]}, 'GLFp': {'d': 2, 'p': 499, 'gens': [2394616189, 61877246503]}, 'Perm': {'d': 108, 'gens': [56213055407243470421616, 12380921884109809718625918504485969512169560791333763137415761458481668540468257551551837625132581751459691578892934601249531595513914854274274549436397754221728366592000000, 701391835124898593479300, 1350198335266800302106816]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [166], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{25}\\times C_{83}', 'transitive_degree': 2075, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '498.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4100, 'aut_gen_orders': [820, 2050, 410, 820], 'aut_gens': [[1, 2], [1495, 1976], [3487, 9712], [3985, 6518], [8965, 4696]], 'aut_group': None, 'aut_hash': 7769306639690861481, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 82000, 'aut_permdeg': 109, 'aut_perms': [125555094366623565765437882654827380035507262165093884116958140064536513332923597928420624737592834670908584044187601868747112877151610943538330313376834743025444534644795291334, 67562577745034697183856093284881101741974165323377147769398082133514383009409343736242459964846061290327632678205115771438778378684513107611285409365491110613023022151584780081, 35100031188454544351036207792202461285912090484518259392278978557086341455519998344241904584140320655254823978900739957878194876561297313212392186228023973002399113583337970440, 37823704201380736101725879268999558316775688890745391484590117892472401574076939718797017787751706675828377634318503626452785405814973846495245473710860300454972215816785443379], 'aut_phi_ratio': 25.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [3, 1, 2, 1], [5, 2, 2, 1], [6, 25, 2, 1], [15, 2, 4, 1], [25, 2, 10, 1], [75, 2, 20, 1], [83, 1, 82, 1], [166, 25, 82, 1], [249, 1, 164, 1], [415, 2, 164, 1], [498, 25, 164, 1], [1245, 2, 328, 1], [2075, 2, 820, 1], [6225, 2, 1640, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{82}\\times D_{25}.C_{10}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 82, 'autcent_group': '164.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 164, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{82}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 100, 'autcentquo_group': '500.18', 'autcentquo_hash': 18, 'autcentquo_nilpotent': False, 'autcentquo_order': 500, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{25}:C_{20}', 'cc_stats': [[1, 1, 1], [2, 25, 1], [3, 1, 2], [5, 2, 2], [6, 25, 2], [15, 2, 4], [25, 2, 10], [75, 2, 20], [83, 1, 82], [166, 25, 82], [249, 1, 164], [415, 2, 164], [498, 25, 164], [1245, 2, 328], [2075, 2, 820], [6225, 2, 1640]], 'center_label': '249.1', 'center_order': 249, 'central_product': True, 'central_quotient': '50.1', 'commutator_count': 1, 'commutator_label': '25.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '3.1', '5.1', '5.1', '83.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['50.1', 1], ['83.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [3, 1, 2, 1], [5, 2, 2, 1], [6, 25, 2, 1], [15, 2, 4, 1], [25, 2, 10, 1], [75, 2, 20, 1], [83, 1, 82, 1], [166, 25, 82, 1], [249, 1, 164, 1], [415, 2, 164, 1], [498, 25, 164, 1], [1245, 2, 328, 1], [2075, 2, 820, 1], [6225, 2, 1640, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1008, 'exponent': 12450, 'exponents_of_order': [2, 1, 1, 1], 'factors_of_aut_order': [2, 5, 41], 'factors_of_order': [2, 3, 5, 83], 'faithful_reps': [[2, 0, 1640]], 'familial': False, 'frattini_label': '5.1', 'frattini_quotient': '2490.c', 'hash': 146291328491806769, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 50, 'inner_gen_orders': [2, 25], 'inner_gens': [[1, 998], [11455, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 50, 'inner_split': False, 'inner_tex': 'D_{25}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': None, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 498], [2, 2988]], 'label': '12450.a', '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': True, 'monomial': True, 'name': 'D25*C249', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 3486, 'number_divisions': 16, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 24, 'number_subgroups': 136, 'old_label': None, 'order': 12450, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 25], [3, 2], [5, 4], [6, 50], [15, 8], [25, 20], [75, 40], [83, 82], [166, 2050], [249, 164], [415, 328], [498, 4100], [1245, 656], [2075, 1640], [6225, 3280]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 410, 'outer_gen_orders': [2, 2, 410], 'outer_gen_pows': [1, 0, 30], 'outer_gens': [[1, 9464], [8467, 4148], [8467, 4522]], 'outer_group': '1640.68', 'outer_hash': 68, 'outer_nilpotent': True, 'outer_order': 1640, 'outer_permdeg': 52, 'outer_perms': [258623241511168180642964355153611979969197632389120000000000, 608540487275778729052895127676448988867522029011599360000000000, 1551118753412443312408155939034436251412140723213229359104000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{410}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 111, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 83], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': None, 'representations': {'PC': {'code': '4817168934281286403694669736312862619', 'gens': [1, 2], 'pres': [5, -2, -3, -5, -5, -83, 9981, 36, 44912, 87, 50403, 118]}, 'GLFp': {'d': 2, 'p': 499, 'gens': [19294591140, 61877246503]}, 'Perm': {'d': 111, 'gens': [16029490886463295123935167576806151581145797066666849879769743752513146011522574985224632792864907282951383336791038891685958597419586661852761536758790555486362336528920420940313, 292746186013923258632440779696207940821736472963558227764261240642692683751598574382439618410335469548986196468085023257632743917097568751776855558061450854400000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [498], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{25}\\times C_{249}', 'transitive_degree': 6225, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
