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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1458.425', 'ambient_counter': 425, 'ambient_order': 1458, 'ambient_tex': 'C_3^5:C_6', 'central': True, 'central_factor': False, 'centralizer_order': 1458, 'characteristic': False, 'core_order': 6, 'counter': 56, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '1458.425.243.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '243.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '243.37', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 37, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 243, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3^4:C_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '6.2', 'subgroup_hash': 2, 'subgroup_order': 6, 'subgroup_tex': 'C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1458.425', 'aut_centralizer_order': 17006112, 'aut_label': '243.b1', 'aut_quo_index': 1, 'aut_stab_index': 9, 'aut_weyl_group': '2.1', 'aut_weyl_index': 153055008, 'centralizer': '1.a1', 'complements': ['6.c1'], 'conjugacy_class_count': 9, 'contained_in': ['81.a1', '81.f1', '81.g1'], 'contains': ['486.b1', '729.a1'], 'core': '243.b1', 'coset_action_label': None, 'count': 9, 'diagramx': [5312, 1543, 5254, 8062], 'generators': [729, 1], 'label': '1458.425.243.b1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '243.b1', 'normal_contained_in': ['81.a1'], 'normal_contains': ['486.b1', '729.a1'], 'normalizer': '1.a1', 'old_label': '243.b1', 'projective_image': '243.37', 'quotient_action_image': '1.1', 'quotient_action_kernel': '243.37', 'quotient_action_kernel_order': 243, 'quotient_fusion': None, 'short_label': '243.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '6.2', '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], [5]], '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], [2, 1, 1, 1], [3, 1, 2, 1], [6, 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], [2, 1, 1], [3, 1, 2], [6, 1, 2]], 'center_label': '6.2', 'center_order': 6, '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', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 6]], 'label': '6.2', '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': 'C6', 'ngens': 2, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 6, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5]], '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': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2]], 'representations': {'PC': {'code': 21, 'gens': [1], 'pres': [2, -2, -3, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [73]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 56]}, 'Perm': {'d': 5, 'gens': [24, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6', 'transitive_degree': 6, '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': '162.55', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 6, 'aut_exponent': 72, 'aut_gen_orders': [6, 12, 6, 18], 'aut_gens': [[1, 3, 9, 27, 81, 243], [650, 492, 188, 1113, 1053, 243], [569, 705, 1200, 1196, 648, 405], [649, 1195, 23, 165, 486, 324], [487, 491, 1039, 1028, 162, 243]], 'aut_group': None, 'aut_hash': 8737133825163724417, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 306110016, 'aut_permdeg': 494, 'aut_perms': 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'aut_phi_ratio': 629856.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 8, 1], [3, 1, 18, 1], [3, 3, 72, 1], [3, 9, 54, 1], [6, 1, 8, 1], [6, 1, 18, 1], [6, 3, 72, 1], [6, 9, 54, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^7.C_3^4.C_3^2.Q_8.D_6.C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': None, 'autcent_hash': 720202149919232562, 'autcent_nilpotent': False, 'autcent_order': 354294, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^7.C_3^4.C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '864.4661', 'autcentquo_hash': 4661, 'autcentquo_nilpotent': False, 'autcentquo_order': 864, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 26], [3, 3, 72], [3, 9, 54], [6, 1, 26], [6, 3, 72], [6, 9, 54]], 'center_label': '54.15', 'center_order': 54, 'central_product': True, 'central_quotient': '27.5', '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', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 425, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['243.37', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 13], [3, 3, 2, 36], [3, 9, 2, 27], [6, 1, 2, 13], [6, 3, 2, 36], [6, 9, 2, 27]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 7800, 'exponent': 6, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '162.55', 'hash': 425, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [1, 3, 3, 3, 1, 1], 'inner_gens': [[1, 3, 9, 27, 81, 243], [1, 3, 495, 27, 81, 243], [1, 975, 9, 108, 81, 243], [1, 3, 171, 27, 81, 243], [1, 3, 9, 27, 81, 243], [1, 3, 9, 27, 81, 243]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 27, 'inner_split': False, 'inner_tex': 'C_3^3', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 162], [3, 144]], 'label': '1458.425', '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': False, 'monomial': True, 'name': 'C3^5:C6', 'ngens': 7, 'nilpotency_class': 2, 'nilpotent': True, '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 306, 'number_divisions': 154, 'number_normal_subgroups': 620, 'number_subgroup_autclasses': 64, 'number_subgroup_classes': 3824, 'number_subgroups': 11528, 'old_label': None, 'order': 1458, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 1], [3, 728], [6, 728]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 72, 'outer_gen_orders': [6, 24, 24, 18], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[569, 520, 1092, 655, 486, 1296], [650, 197, 14, 224, 648, 1296], [164, 116, 199, 1195, 648, 1296], [1136, 1189, 693, 192, 648, 810]], 'outer_group': None, 'outer_hash': 7594568951909758545, 'outer_nilpotent': False, 'outer_order': 11337408, 'outer_permdeg': 63, 'outer_perms': [913898520988183380515054143792053262989622756177164171737168363814074442539034257843265, 410234835866228515507471888604290056043393142620651320247562652433553323996454019783209, 473783883020637954712850417849068141050205566294539794669446332149522958545506109029766, 946880262153562672781850738464996763065142974111425770983206065278705065816581218919671], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_3^6.C_3^4.Q_8.D_6.C_2', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3, 3, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 80], [6, 72]], 'representations': {'PC': {'code': 219147150062784637, 'gens': [1, 2, 3, 4, 5, 6], 'pres': [7, -3, -3, -3, -3, -3, -2, -3, 3474, 353, 124]}, 'Perm': {'d': 23, 'gens': [250845943359524868, 2812067596232294400, 79890976, 43545600, 5129431175169100800, 93313, 1124000727777607680000]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^5:C_6', 'transitive_degree': 162, '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': '27.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 72, 'aut_gen_orders': [2, 6, 12, 12, 3, 3, 3, 3, 3, 3], 'aut_gens': [[24577758, 24507118, 1305911, 2939606, 36731395], [20201582, 20254942, 1305911, 2939606, 32428289], [37043725, 7046280, 4733462, 32864349, 2794104], [32864349, 14633333, 5611474, 24577758, 6973453], [11495158, 24898809, 1359298, 43014227, 20327769], [24577758, 24507118, 27715450, 2939606, 36731395], [24577758, 24507118, 23854965, 2939606, 36731395], [24577758, 20327769, 31289712, 2939606, 14686477], [24577758, 37169915, 5504967, 2939606, 7357854], [24577758, 24507118, 31289712, 2939606, 36731395], [24577758, 24507118, 5504967, 2939606, 36731395]], 'aut_group': '629856.le', 'aut_hash': 4095539607023997171, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 629856, 'aut_permdeg': 81, 'aut_perms': [4438536991289698901915356563578909183685878414777360979229137984440562671025735739767449666153971292096149085749967681875, 5085518333382388263346035390932792195270973485023344467110358527419924912953707193938958285813220878302567044794628062233, 615942507436528910147344144006332404692093156457902348345707795575727499063546661619958579594708483908970627531721902613, 4482997165595264759074241758044633386017280260732316528643563563025087387260588873608186815592099148501517411354784642529, 4024520134362269993714032188676026536797626722731017805488974739124359354432590639057130287787761841182626808004736515913, 3325857953924182723175397805874612611501147124510928170903577887831533295019948625398266304766588382118821791141330690354, 1296506529179032578018496511803457939280506031162943861893353101904474499555905162850885385519383886642327827600413510400, 237255623980108845294249274947844995836080539069233676444411718834230511741354698245078772909072741519348245496537004800, 1525721429271618225767633086240326656933682267939565599704719719427575010368014799997970628638076167504919330048541348976, 366952014894226021163661737201074106392785377583120419939421282090602069362024653547293263214687824294495674043901625277], 'aut_phi_ratio': 3888.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 8, 1], [3, 3, 24, 1], [3, 9, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^5.(S_3\\times C_3^2:\\GL(2,3))', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '729.504', 'autcent_hash': 504, 'autcent_nilpotent': True, 'autcent_order': 729, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '864.4661', 'autcentquo_hash': 4661, 'autcentquo_nilpotent': False, 'autcentquo_order': 864, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [3, 1, 8], [3, 3, 24], [3, 9, 18]], 'center_label': '9.2', 'center_order': 9, 'central_product': False, 'central_quotient': '27.5', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 37, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4], [3, 3, 2, 12], [3, 9, 2, 9]], 'element_repr_type': 'GLFp', 'elementary': 3, 'eulerian_function': 13, 'exponent': 3, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '27.5', 'hash': 37, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [1, 3, 3, 1, 3], 'inner_gens': [[24577758, 24507118, 1305911, 2939606, 36731395], [24577758, 24507118, 5504967, 2939606, 36731395], [24577758, 20327769, 1305911, 2939606, 11537230], [24577758, 24507118, 1305911, 2939606, 36731395], [24577758, 24507118, 41439590, 2939606, 36731395]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 27, 'inner_split': False, 'inner_tex': 'C_3^3', 'inner_used': [2, 3, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 27], [3, 24]], 'label': '243.37', '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': False, 'monomial': True, 'name': 'C3^4:C3', 'ngens': 5, 'nilpotency_class': 2, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 51, 'number_divisions': 26, 'number_normal_subgroups': 45, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 189, 'number_subgroups': 531, 'old_label': None, 'order': 243, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 242]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 72, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [14408200, 14408200], 'outer_gens': [[11495158, 20699780, 17565602, 32864349, 7192025], [20201582, 24507118, 8679975, 43014227, 24783208]], 'outer_group': '23328.cj', 'outer_hash': 2313055496293256831, 'outer_nilpotent': False, 'outer_order': 23328, 'outer_permdeg': 18, 'outer_perms': [47270795755726, 1570542346920740], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_3^5:(C_2\\times \\GL(2,3))', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 13], [6, 12]], 'representations': {'PC': {'code': 4079888, 'gens': [1, 2, 3, 4, 5], 'pres': [5, -3, 3, 3, -3, 3, 571, 457]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 24507118, 1305911, 2939606, 36731395]}, 'Perm': {'d': 18, 'gens': [26259825726468, 422466042758176, 801967071628800, 1114594614698113, 1114594614604800]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:C_3', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
