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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1458.1369', 'ambient_counter': 1369, 'ambient_order': 1458, 'ambient_tex': '(C_3\\times C_9^2):C_6', 'central': False, 'central_factor': False, 'centralizer_order': 243, 'characteristic': True, 'core_order': 27, 'counter': 69, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1458.1369.54.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '54.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '54.5', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 54, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3^2:C_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '27.5', 'subgroup_hash': 5, 'subgroup_order': 27, 'subgroup_tex': 'C_3^3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1458.1369', 'aut_centralizer_order': 2187, 'aut_label': '54.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '36.12', 'aut_weyl_index': 2187, 'centralizer': '6.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['18.a1.a1', '18.e1.a1', '18.f1.a1', '18.g1.a1', '27.f1.a1'], 'contains': ['162.a1.a1', '162.d1.a1', '162.k1.a1', '162.k1.b1', '162.k1.c1'], 'core': '54.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3665, 2383, 7226, 5706, 6704, 5958, 4645, 7773], 'generators': [54, 18, 486], 'label': '1458.1369.54.a1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '54.a1.a1', 'normal_contained_in': ['18.a1.a1'], 'normal_contains': ['162.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '54.a1.a1', 'projective_image': '1458.1369', 'quotient_action_image': '6.2', 'quotient_action_kernel': '9.2', 'quotient_action_kernel_order': 9, 'quotient_fusion': None, 'short_label': '54.a1.a1', 'subgroup_fusion': None, 'weyl_group': '6.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '27.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 312, 'aut_gen_orders': [2, 13], 'aut_gens': [[1, 3, 9], [6, 2, 9], [5, 17, 11]], 'aut_group': '11232.a', 'aut_hash': 778507202365856770, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11232, 'aut_permdeg': 15, 'aut_perms': [6452863345, 297839648544], 'aut_phi_ratio': 624.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [3, 1, 26, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(3,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 312, 'autcent_group': '11232.a', 'autcent_hash': 778507202365856770, 'autcent_nilpotent': False, 'autcent_order': 11232, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(3,3)', '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, 26]], 'center_label': '27.5', 'center_order': 27, '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': ['3.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 13]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '27.5', 'hash': 5, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 3, 9], [1, 3, 9], [1, 3, 9]], '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, 27]], 'label': '27.5', 'linC_count': 1872, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 234, 'linQ_dim': 6, 'linQ_dim_count': 234, 'linR_count': 234, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^3', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 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': 27, 'number_divisions': 14, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 28, 'number_subgroups': 28, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 26]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 312, 'outer_gen_orders': [2, 13], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 2, 9], [5, 17, 11]], 'outer_group': '11232.a', 'outer_hash': 778507202365856770, 'outer_nilpotent': False, 'outer_order': 11232, 'outer_permdeg': 15, 'outer_perms': [6452863345, 297839648544], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(3,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 13]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -3, 3, 3]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41624334336939899, 125101725607167719, 125101750005091125]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [10475581, 9534373, 23068812]}, 'Perm': {'d': 9, 'gens': [80640, 240, 4]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3', 'transitive_degree': 27, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [6, 6, 18], 'aut_gens': [[1, 6, 54, 162], [323, 528, 1044, 1446], [925, 30, 144, 810], [659, 978, 540, 480]], 'aut_group': None, 'aut_hash': 5248920786018732718, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 78732, 'aut_permdeg': 297, 'aut_perms': [5689147093264135390820841173422413875480723735166448164098474191190671089767141628569731560152117453352824579706746534890537042455338701286579742705748228175774704442353595731847484645379636907677867949056041508661881502948998875327725017733730208475943918219631409058070189625082763875990323434982418958555042599838695624072948381062232108311883934271923697090226158171492727770297781624338375570855534924016420979359522550434959447888343945741216888931049157505724321773283912472415533291001696431297239839207176982217460477571182097023118320855716571799703209162568735829031610934162021645416652495154938, 10425698917876422479158845488234958711412883952984084460695807179783716710735977215640965759634112720713493645009143624118019874999967757683641678258269153417529409119984187299330981588550717917895834056123502807331669826328496761339329044463520822218581756505571482518580534938256428702505353374933322669800191298468021286475011279609225972051499987628746466136553170153624274675798646461813016436142871272715973288520257172909873609200116407855874178270470900456245608376152558400835121869783660806111913710331699318868503289325157269261044564444574987477419950460845905018343249251631701157210230880918061, 8110315459421742739987144287980652618271459968140604878085682554785928944048992047472060035206350864175727389260449982269677756331061754742117131066413004893152092798365954116012932403223672604551168831207028158610053621174164002543253835248761933330812068073825187350082951986397616326911916483512274637835250773317484539557007305068916807966913736735571027325373170779233692743189195433391358126357517293931030645987498392391844279539055361379554152995312159626681851101682445099405223843614651399493867757080490384264911284269953898336554559046675117050159243463028898583164268854995762754258653327320369], 'aut_phi_ratio': 162.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 6, 3, 1], [3, 27, 2, 1], [3, 54, 2, 1], [6, 243, 2, 1], [9, 2, 3, 1], [9, 6, 2, 1], [9, 6, 3, 2], [9, 6, 9, 1], [9, 6, 18, 1], [9, 54, 2, 3]], 'aut_supersolvable': True, 'aut_tex': 'C_9^2.C_3^5.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': None, 'autcentquo_hash': 5248920786018732718, 'autcentquo_nilpotent': False, 'autcentquo_order': 78732, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': '\\He_3:C_3.C_3^5.C_2^2', 'cc_stats': [[1, 1, 1], [2, 243, 1], [3, 2, 1], [3, 6, 4], [3, 27, 2], [3, 54, 2], [6, 243, 2], [9, 2, 3], [9, 6, 35], [9, 54, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1458.1369', 'commutator_count': 1, 'commutator_label': '243.31', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1369, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 4], [3, 27, 2, 1], [3, 54, 2, 1], [6, 243, 2, 1], [9, 2, 3, 1], [9, 6, 1, 2], [9, 6, 3, 11], [9, 54, 2, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 19656, 'exponent': 18, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 1, 27]], 'familial': False, 'frattini_label': '27.2', 'frattini_quotient': '54.13', 'hash': 1369, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 9, 3, 9], 'inner_gens': [[1, 516, 1098, 354], [1003, 6, 54, 162], [631, 6, 54, 162], [1321, 6, 54, 162]], 'inner_hash': 1369, 'inner_nilpotent': False, 'inner_order': 1458, 'inner_split': True, 'inner_tex': '(C_3\\times C_9^2):C_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 6, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 6, 'irrep_stats': [[1, 6], [2, 12], [6, 39]], 'label': '1458.1369', 'linC_count': 27, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 9, 'linQ_dim': 18, 'linQ_dim_count': 9, 'linR_count': 27, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C3*C9^2):C6', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 17, 'number_characteristic_subgroups': 17, 'number_conjugacy_classes': 57, 'number_divisions': 27, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 102, 'number_subgroup_classes': 157, 'number_subgroups': 4026, 'old_label': None, 'order': 1458, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 243], [3, 188], [6, 486], [9, 540]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 6, 3], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 42, 72, 1152], [5, 498, 1080, 744], [1, 6, 54, 198]], 'outer_group': '54.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 54, 'outer_permdeg': 9, 'outer_perms': [80640, 145, 3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times C_3^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [4, 4], [6, 3], [18, 12]], 'representations': {'PC': {'code': 58875747788122250164034138939414236438081819327, 'gens': [1, 3, 5, 6], 'pres': [7, 2, 3, 3, 3, 3, 3, 3, 14, 10838, 5364, 79, 1011, 38434, 18596, 14873, 4422, 166, 49398, 12802]}, 'Perm': {'d': 27, 'gens': [15618145818219673679602412, 7550102686338507196661760000, 2644117572280194, 1258350444907174137666609094, 2935341388602599905184532724, 711549292653507, 806634631153204606767248884]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_3\\times C_9^2):C_6', 'transitive_degree': 27, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [3, 3, 2, 3, 2], 'aut_gens': [[1, 6, 18], [7, 6, 18], [1, 24, 18], [43, 12, 36], [19, 6, 18], [23, 24, 36]], 'aut_group': '108.17', 'aut_hash': 17, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 108, 'aut_permdeg': 9, 'aut_perms': [192914, 98790, 221213, 93877, 90741], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [6, 9, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2:D_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '108.17', 'autcentquo_hash': 17, 'autcentquo_nilpotent': False, 'autcentquo_order': 108, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2:D_6', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 1], [3, 3, 2], [3, 6, 3], [6, 9, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '54.5', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [6, 9, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '18.3', 'hash': 5, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [6, 3, 3], 'inner_gens': [[1, 48, 36], [31, 6, 18], [37, 6, 18]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 54, 'inner_split': True, 'inner_tex': 'C_3^2:C_6', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 6], [2, 3], [6, 1]], 'label': '54.5', 'linC_count': 1, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:C6', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 10, 'number_divisions': 7, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 54, 'old_label': None, 'order': 54, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 9], [3, 26], [6, 18]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5, 48, 18]], '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': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [6, 1]], 'representations': {'PC': {'code': 2606065431, 'gens': [1, 3, 4], 'pres': [4, -2, -3, -3, -3, 8, 578, 258, 579]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125124615946045039, 78750815771418988]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 11017, 14041, 8101]}, 'Perm': {'d': 9, 'gens': [5065, 276480, 243, 80884]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:C_6', 'transitive_degree': 9, 'wreath_data': None, 'wreath_product': False}
