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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1134.198', 'ambient_counter': 198, 'ambient_order': 1134, 'ambient_tex': 'C_{42}:\\He_3', 'central': False, 'central_factor': False, 'centralizer_order': 378, 'characteristic': True, 'core_order': 21, 'counter': 31, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1134.198.54.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '54.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '54.15', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 15, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 54, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3^2\\times C_6', 'simple': False, 'solvable': True, 'special_labels': ['D', 'L1', 'D1', 'C4'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '21.2', 'subgroup_hash': 2, 'subgroup_order': 21, 'subgroup_tex': 'C_{21}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1134.198', 'aut_centralizer_order': 10206, 'aut_label': '54.a1', 'aut_quo_index': 104, 'aut_stab_index': 1, 'aut_weyl_group': '12.5', 'aut_weyl_index': 10206, 'centralizer': '3.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['18.a1', '18.b1', '18.c1', '27.a1'], 'contains': ['162.a1', '378.a1'], 'core': '54.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6011, 4903, 4080, 5662], 'generators': [9, 162], 'label': '1134.198.54.a1', 'mobius_quo': 1, 'mobius_sub': 27, 'normal_closure': '54.a1', 'normal_contained_in': ['18.a1', '18.b1', '18.c1', '27.a1'], 'normal_contains': ['162.a1', '378.a1'], 'normalizer': '1.a1', 'old_label': '54.a1', 'projective_image': '378.52', 'quotient_action_image': '3.1', 'quotient_action_kernel': '18.5', 'quotient_action_kernel_order': 18, 'quotient_fusion': None, 'short_label': '54.a1', 'subgroup_fusion': None, 'weyl_group': '3.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '21.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1], [13], [11]], 'aut_group': '12.5', 'aut_hash': 5, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 12, 'aut_permdeg': 7, 'aut_perms': [24, 723], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [7, 1, 6, 1], [21, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_6', '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], [7, 1, 6], [21, 1, 12]], 'center_label': '21.2', 'center_order': 21, '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', '7.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [7, 1, 6, 1], [21, 1, 12, 1]], 'element_repr_type': 'PC', 'elementary': 21, 'eulerian_function': 1, 'exponent': 21, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3, 7], 'faithful_reps': [[1, 0, 12]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '21.2', 'hash': 2, 'hyperelementary': 21, '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': 12, 'irrQ_dim': 12, 'irrR_degree': 2, 'irrep_stats': [[1, 21]], 'label': '21.2', 'linC_count': 12, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C21', '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': 21, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 21, 'order_factorization_type': 11, 'order_stats': [[1, 1], [3, 2], [7, 6], [21, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[13], [11]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [24, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [3, 7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [6, 1], [12, 1]], 'representations': {'PC': {'code': 191, 'gens': [1], 'pres': [2, -3, -7, 6]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [351, 1376]}, 'Perm': {'d': 10, 'gens': [725760, 4320]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [21], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{21}', 'transitive_degree': 21, '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': '54.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 126, 'aut_gen_orders': [6, 6, 6, 6], 'aut_gens': [[1, 3, 9, 27], [19, 490, 18, 631], [20, 777, 18, 802], [10, 21, 9, 360], [2, 706, 9, 857]], 'aut_group': None, 'aut_hash': 5863880736905008192, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 122472, 'aut_permdeg': 191, 'aut_perms': [1441041035729374478940225965930042393387661923569930811044810861589281907165995304852857845720170272429801635286553069078918794835222269274917149218418229531538563569172284945070496689107113779659987469745907503034602040928764587550155890836089185975149364364986450852057228324927792283024066376000897415459985663796942662989492730805482885617988856896378, 842272596515551016822468944638709987527677257078412934284858442373098931794839862692786369153993005470378673940573567324608018097991228542488487039895237241360437341138084290802476220482170210056810452599466224365059625767889383384503779962441751820075981881105293185939476867351363669587830897167939176010387938175363609477148259727941670586737810029901, 1150954336270502087078401063302739507657373988852816799937150625988862972744420728874604087792276983876075502362770219460270979460674868594907511922761885309420404894544021200822790176260186798425854525250460087739175752809947179498572040405604279542757502622588952358344247095470908731626967874669184564622280572500428805746061558614341081441384679598514, 149513167718566778636613032659140103242267389141435617839996576439521235121808555112457911942262849860263528643392416622686757929982164684286673103000549530750397147190389270332820283551113728560583041987866084837370241336472740255583093685707181638892088707714609818589395885172671590042571839788127212367725696637022878325424843779589645001324161012030], 'aut_phi_ratio': 378.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [3, 3, 6, 1], [3, 21, 9, 2], [6, 1, 2, 1], [6, 1, 6, 1], [6, 3, 6, 1], [6, 21, 9, 2], [7, 3, 2, 1], [14, 3, 2, 1], [21, 3, 4, 1], [21, 3, 12, 1], [21, 3, 36, 1], [42, 3, 4, 1], [42, 3, 12, 1], [42, 3, 36, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^4.S_3^2\\times F_7', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '486.183', 'autcent_hash': 183, 'autcent_nilpotent': False, 'autcent_order': 486, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^4:S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 8], [3, 3, 6], [3, 21, 18], [6, 1, 8], [6, 3, 6], [6, 21, 18], [7, 3, 2], [14, 3, 2], [21, 3, 52], [42, 3, 52]], 'center_label': '18.5', 'center_order': 18, 'central_product': True, 'central_quotient': '63.3', 'commutator_count': 1, 'commutator_label': '21.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 198, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['189.8', 1], ['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [3, 3, 2, 3], [3, 21, 2, 9], [6, 1, 2, 4], [6, 3, 2, 3], [6, 21, 2, 9], [7, 3, 2, 1], [14, 3, 2, 1], [21, 3, 4, 4], [21, 3, 12, 3], [42, 3, 4, 4], [42, 3, 12, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5824, 'exponent': 42, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '378.52', 'hash': 198, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 21, 'inner_gen_orders': [1, 3, 1, 21], 'inner_gens': [[1, 3, 9, 27], [1, 3, 9, 693], [1, 3, 9, 27], [1, 498, 9, 27]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 63, 'inner_split': False, 'inner_tex': 'C_{21}:C_3', 'inner_used': [2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 54], [3, 120]], 'label': '1134.198', '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': 'C42:He3', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 174, 'number_divisions': 50, 'number_normal_subgroups': 84, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 224, 'number_subgroups': 1328, 'old_label': None, 'order': 1134, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [3, 404], [6, 404], [7, 6], [14, 6], [21, 156], [42, 156]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [3, 2, 3, 6, 6, 2, 6, 6], 'outer_gen_pows': [0, 0, 0, 0, 0, 6, 0, 0], 'outer_gens': [[1, 14, 9, 36], [2, 768, 18, 783], [1, 3, 9, 45], [1, 22, 9, 847], [1, 759, 9, 522], [1, 3, 9, 855], [1, 779, 18, 783], [19, 390, 9, 522]], 'outer_group': '1944.2710', 'outer_hash': 2710, 'outer_nilpotent': False, 'outer_order': 1944, 'outer_permdeg': 20, 'outer_perms': [379586491085297544, 34235089789824000, 8792166, 38440576314484561, 898487379093427201, 1, 398078314790612688, 506964703687369201], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_3^3:S_3^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 26], [6, 8], [12, 8], [36, 6]], 'representations': {'PC': {'code': 245779461479106119416403, 'gens': [1, 2, 3, 4], 'pres': [6, 3, 3, 3, 2, 3, 7, 5553, 69, 2260, 118, 7787]}, 'GLZN': {'d': 2, 'p': 63, 'gens': [1000189, 6251200, 15502976, 250615, 251371, 336967]}, 'Perm': {'d': 21, 'gens': [1, 128089326405201888, 6714914766, 11090880, 8892366, 2689751484516096000]}}, 'schur_multiplier': [3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{42}:\\He_3', 'transitive_degree': 378, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '54.15', '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], [36, 22, 28], [44, 7, 50]], 'aut_group': '11232.a', 'aut_hash': 778507202365856770, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11232, 'aut_permdeg': 15, 'aut_perms': [965974769425, 438185353320], 'aut_phi_ratio': 624.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 26, 1], [6, 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], [2, 1, 1], [3, 1, 26], [6, 1, 26]], 'center_label': '54.15', 'center_order': 54, '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', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 13], [6, 1, 2, 13]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 7, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '54.15', 'hash': 15, '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, 54]], 'label': '54.15', 'linC_count': 13104, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1638, 'linQ_dim': 6, 'linQ_dim_count': 1638, 'linR_count': 1638, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2*C6', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 54, 'number_divisions': 28, 'number_normal_subgroups': 56, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 56, 'number_subgroups': 56, 'old_label': None, 'order': 54, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [3, 26], [6, 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': [[36, 22, 28], [44, 7, 50]], 'outer_group': '11232.a', 'outer_hash': 778507202365856770, 'outer_nilpotent': False, 'outer_order': 11232, 'outer_permdeg': 15, 'outer_perms': [965974769425, 438185353320], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(3,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 26]], 'representations': {'PC': {'code': 8269, 'gens': [1, 2, 3], 'pres': [4, -3, -3, -2, -3, 34]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41624334336939899, 125101750005088939, 24992888769124995]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [10475581, 28836015, 22014775, 23068812]}, 'Perm': {'d': 11, 'gens': [3628800, 80640, 240, 4]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 6], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2\\times C_6', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
