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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '37044.e', 'ambient_counter': 5, 'ambient_order': 37044, 'ambient_tex': '\\He_7:(C_3^2\\times D_6)', 'central': False, 'central_factor': False, 'centralizer_order': 126, 'characteristic': False, 'core_order': 7, 'counter': 170, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '37044.e.1764.i1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '1764.i1', 'outer_equivalence': True, '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': 1764, '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': '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': '37044.e', 'aut_centralizer_order': None, 'aut_label': '1764.i1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '294.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['252.f1', '588.a1', '588.l1', '882.d1'], 'contains': ['5292.a1', '12348.d1'], 'core': '5292.a1', 'coset_action_label': None, 'count': 98, 'diagramx': [3340, -1, 4363, -1], 'generators': [16807270434870, 14230002226939], 'label': '37044.e.1764.i1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '98.a1', 'old_label': '1764.i1', 'projective_image': '37044.e', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1764.i1', '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': '36.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 84, 'aut_gen_orders': [14, 14, 6, 12], 'aut_gens': [[9136408354495, 9567391489514, 11810182508209, 21940152462727, 9495688004000, 31584908075045, 7394473631462, 28186167033742], [19125156069853, 3071306945005, 23337298129670, 9353313845575, 18991376008000, 30937847535514, 18140428511610, 20081273301458], [27943184661968, 8066376848598, 24773747668640, 24796747146699, 18991376008000, 4872587944260, 22047655452128, 28186167033742], [20854056837171, 7123095234355, 24766649562356, 28654602347963, 18991376008000, 4393599959002, 11454005438155, 1487683265364], [20702030637967, 14690582783695, 4668081670517, 25671161683103, 18991376008000, 30152601068257, 8834902986995, 13544849055836]], 'aut_group': None, 'aut_hash': 1912888100653018415, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 148176, 'aut_permdeg': 588, 'aut_perms': 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'aut_phi_ratio': 14.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 49, 1, 1], [2, 147, 2, 1], [3, 1, 2, 1], [3, 98, 1, 1], [3, 98, 2, 1], [3, 98, 3, 2], [3, 343, 3, 2], [6, 49, 2, 1], [6, 98, 1, 1], [6, 98, 2, 1], [6, 147, 4, 1], [6, 343, 3, 2], [6, 686, 3, 2], [6, 1029, 6, 2], [7, 6, 1, 1], [7, 84, 1, 1], [7, 126, 2, 1], [14, 294, 1, 1], [14, 882, 2, 1], [21, 6, 2, 1], [21, 84, 2, 1], [21, 126, 4, 1], [21, 294, 2, 1], [21, 294, 4, 1], [21, 588, 3, 2], [42, 294, 2, 2], [42, 294, 4, 1], [42, 882, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '\\He_7.(C_6\\times S_3^2).C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '24696.h', 'autcentquo_hash': 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1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 49, 1, 1], [2, 147, 1, 2], [3, 1, 2, 1], [3, 98, 1, 1], [3, 98, 2, 4], [3, 343, 2, 3], [6, 49, 2, 1], [6, 98, 1, 1], [6, 98, 2, 1], [6, 147, 2, 2], [6, 343, 2, 3], [6, 686, 2, 3], [6, 1029, 2, 6], [7, 6, 1, 1], [7, 84, 1, 1], [7, 126, 1, 2], [14, 294, 1, 1], [14, 882, 1, 2], [21, 6, 2, 1], [21, 84, 2, 1], [21, 126, 2, 2], [21, 294, 2, 1], [21, 294, 4, 1], [21, 588, 2, 3], [42, 294, 2, 2], [42, 294, 4, 1], [42, 882, 2, 2]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 42, 'exponents_of_order': [3, 3, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[42, 0, 12]], 'familial': False, 'frattini_label': '7.1', 'frattini_quotient': '5292.n', 'hash': 7855798439795622267, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [3, 21, 14, 6, 1, 7, 7, 7], 'inner_gens': [[9136408354495, 31987721837770, 16405770270200, 21940152462727, 9495688004000, 13018890495555, 15897285792898, 16807270434870], [2177346105515, 9567391489514, 30537606232365, 19980894921572, 9495688004000, 23822349130192, 27044744357397, 28186167033742], [17765321726969, 28211616471470, 11810182508209, 6632225060598, 9495688004000, 16931726136737, 11454005438155, 20081273301458], [9136408354495, 31857582747939, 9717366629131, 21940152462727, 9495688004000, 16405766752913, 3224327410054, 13544849055836], [9136408354495, 9567391489514, 11810182508209, 21940152462727, 9495688004000, 31584908075045, 7394473631462, 28186167033742], [19567949788913, 31970989164783, 27044744510117, 22217201853311, 9495688004000, 31584908075045, 25307817031564, 28186167033742], [31231305933848, 5159387320453, 4393600935216, 332310625717, 9495688004000, 12894152318439, 7394473631462, 28186167033742], [27728263067888, 9567391489514, 15084185237940, 18678014399348, 9495688004000, 31584908075045, 7394473631462, 28186167033742]], 'inner_hash': 3676818443602011658, 'inner_nilpotent': False, 'inner_order': 12348, 'inner_split': True, 'inner_tex': '\\He_7:(C_6\\times S_3)', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 42, 'irrQ_degree': 84, 'irrQ_dim': 84, 'irrR_degree': 84, 'irrep_stats': [[1, 36], [2, 18], [12, 9], [18, 12], [42, 18]], 'label': '37044.e', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'He7:(C3^2*D6)', 'ngens': 8, '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, 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': 36, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 93, 'number_divisions': 51, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 216, 'number_subgroup_classes': 460, 'number_subgroups': 53984, 'old_label': None, 'order': 37044, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 343], [3, 2942], [6, 19502], [7, 342], [14, 2058], [21, 5976], [42, 5880]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [4747844002000, 4747844002000], 'outer_gens': [[13523237384241, 14570537226190, 5796127193804, 5897835401518, 18991376008000, 5398541562940, 14838904351914, 28186167033742], [21523801175276, 4676780755490, 17716154452380, 5897835401518, 9495688004000, 30661082028272, 7640036134130, 20081273301458]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 49], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 101, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 8], [12, 1], [18, 4], [24, 4], [36, 4], [42, 2], [84, 4], [168, 2]], 'representations': {'PC': {'code': '190652541535688927821789494161321337785985802547135214911279728389233704834812699521218307492187830389200427137907051937491620327935', 'gens': [1, 3, 5, 7, 8], 'pres': [8, -2, -3, -2, -3, 3, -7, 7, -7, 16, 386354, 229474, 66, 3843, 534155, 1450564, 320412, 78020, 12028, 156, 1596677, 10381, 256629, 43229, 1493862, 973742, 21190, 7086, 51782, 5926, 1016071, 338703, 112919, 112927, 18855]}, 'GLFp': {'d': 4, 'p': 7, 'gens': [9136408354495, 9567391489514, 11810182508209, 21940152462727, 9495688004000, 31584908075045, 7394473631462, 28186167033742]}, 'Perm': {'d': 101, 'gens': [98097202876273071242932883012828361348099459168994283045668001988377704000159042113240171848721215384490857431169648444283160096404164583047139701336511740064, 192366300814728093582468831107058254699291474516877634400641867595784459630430184698508327804359364900407515596642520758706552368013726145891350586807608875664, 286635587109982529534707311611665255989106039518602675801083123653954514989956221518372525718972964599754019815509336467670939096654567930136665001459335002480, 7599714176795627378165009276597172894167950141741256117417384244208723743721357332838043744277017177269362236260749287896244568158289982443813102238070425920, 3, 381837950177928444746192472352594381684555244609756523394444934505160657605089913027859578780170432023030723750178550843560358082874692542312629979996930249360, 9475586063638613970331949186488481722708622576335774921901405938261471667142670528381108178317545651868482132619731552388867142761139581995039720897253828228, 477049641826004280740007945659976971528416356327783137218138090761687049166325774246751564675654151623695198764629403642266808067026464136168494021007179345920]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 6], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\He_7:(C_3^2\\times D_6)', 'transitive_degree': 294, 'wreath_data': None, 'wreath_product': False}
