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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '23328.du', 'ambient_counter': 99, 'ambient_order': 23328, 'ambient_tex': 'C_3^3:S_3^2.S_4', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 3, 'counter': 367, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '23328.du.216.ba1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '216.ba1.a1', 'outer_equivalence': False, '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': 216, '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': '108.38', 'subgroup_hash': 38, 'subgroup_order': 108, 'subgroup_tex': 'C_3\\times S_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '23328.du', 'aut_centralizer_order': None, 'aut_label': '216.ba1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '7776.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['72.o1.a1', '72.u1.a1', '72.u1.b1', '108.d1.a1'], 'contains': ['432.p1.a1', '432.bh1.a1', '432.bh1.b1', '648.x1.a1', '648.x1.b1', '648.bi1.a1'], 'core': '7776.a1.a1', 'coset_action_label': None, 'count': 108, 'diagramx': [3538, -1, 3500, -1, 3993, -1, 5480, -1], 'generators': [806634631153204606767248884, 2516538717432286673398548480, 711549292653507, 108062211899663342133718, 3778810052089204], 'label': '23328.du.216.ba1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '108.d1.a1', 'old_label': '216.ba1.a1', 'projective_image': '23328.du', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '216.ba1.a1', 'subgroup_fusion': None, 'weyl_group': '72.46'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 6, 4, 2], 'aut_gens': [[1, 6, 36], [13, 30, 36], [13, 6, 36], [5, 6, 36], [29, 42, 36], [22, 63, 12], [17, 30, 72]], 'aut_group': '144.186', 'aut_hash': 186, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 144, 'aut_permdeg': 8, 'aut_perms': [10800, 11064, 1, 22375, 7566, 10855], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 2, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 2, 1], [3, 2, 4, 1], [3, 4, 1, 1], [3, 4, 2, 1], [6, 3, 4, 1], [6, 6, 2, 1], [6, 6, 4, 1], [6, 9, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3^2:C_2^2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '72.40', 'autcentquo_hash': 40, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)', 'cc_stats': [[1, 1, 1], [2, 3, 2], [2, 9, 1], [3, 1, 2], [3, 2, 6], [3, 4, 3], [6, 3, 4], [6, 6, 6], [6, 9, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '36.10', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 38, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 2], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 1, 2], [3, 2, 2, 2], [3, 4, 1, 1], [3, 4, 2, 1], [6, 3, 2, 2], [6, 6, 1, 2], [6, 6, 2, 2], [6, 9, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '108.38', 'hash': 38, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 3], 'inner_gens': [[1, 30, 36], [13, 6, 72], [1, 78, 36]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'S_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 12], [2, 12], [4, 3]], 'label': '108.38', 'linC_count': 34, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 28, 'linQ_dim': 6, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3*S3^2', 'ngens': 5, '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': 12, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 27, 'number_divisions': 18, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 54, 'number_subgroups': 152, 'old_label': None, 'order': 108, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 15], [3, 26], [6, 66]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[22, 75, 12], [5, 30, 36]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 5], [8, 1]], 'representations': {'PC': {'code': 73347030453419, 'gens': [1, 3, 5], 'pres': [5, -2, -3, -2, -3, -3, 10, 452, 42, 483, 314]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [108470304437280604, 125101732571169122]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [11495158, 22274384, 5504967, 25627796, 20643250]}, 'Perm': {'d': 9, 'gens': [1, 42000, 97224, 136224, 3]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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 S_3^2', 'transitive_degree': 12, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 6, 'aut_exponent': 72, 'aut_gen_orders': [6, 6, 4, 4, 3, 6, 3, 3, 3], 'aut_gens': [[822419622445341580929239305, 8356766692473337207873663173], [2920450606726856674650300845, 7550183767664110684862903734], [403860182441780655992102396, 1258807576303316274857758674], [6694826317899993940425010559, 7098467912684568758389690947], [822525406433157648995123881, 10873224328581532259074373763], [9211229831024606930771761226, 8808423751644186057527543634], [2920099307713695329135356922, 7968957061547591295407683653], [8371898322429249486853349161, 8356793665935545561372866294], [10888382931734464757086590269, 10873305409902222901893569283], [403668652998337115408051885, 8356766692473337207873663173]], 'aut_group': '46656.jt', 'aut_hash': 7932973419676711738, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 46656, 'aut_permdeg': 162, 'aut_perms': [10451432316119256329985288695271173147762491040230726970771441726900173292831180671336258946641590843237876565199123366816531483734024084043644137277005744092239917773783699010361852309957028648311462662715660317968839740758334066497602385538469695605083574565424745596021985981510480939301, 27033216342474169612061301885568823643283107183172732392782647589897047965316235955636612755476011598325462957885783412990129533596438081742191986808351894589029682047809116296507023821177860881254082085964877352661320137624049024628211403442624841136808548179648765976931659356111479154, 8314175316650627995998614369519580840824622649434245833795155765329797491000420307320468360646740448614741925079516960276261738067709642782437431478166884454289802228041590563647244063980463510728415357594783615274500376951838197943831466866052671925194904714452894140945971400840660404433, 10123086359855337214507676577103653631801989142062209615133576181520747663193306668044675381533014347050738978692225614880507092020988456672437500696529926150342433121441483436180407492398810295900957341582452407222903238061791834815808875776885166558117763027885873773917449921597093599252, 9635815988893364091134805942317226167734874543951861969989742828911595399854436565903135921518331239129716859306218963684496707802792848187375709508633678433820559455163469922904622278299381000786852573726330563790359106217467931899550978587122345791126587276133651996040965656713427702542, 11687609757161874781490478079049154359021049749509998828515258924095977474726402386159024285130196556807396009014743541185768150073898483806297858586775832750222467170075528105574579699738994119225468284832013947612831359356289789978568396725351800720735110671568010850367406536521387162665, 9136449099391063500096059649496110213288528740448990234893458238654391949271677117461698477611347675948029055282683033479863819307570464743040031462465876023254723111970506419018078786057694800273371667935320285993399669065294328661322344183104764472970889775793951449618741106564821487747, 1369956327942788860678637661996485421990960919303906407169499963551600173654712714750783659066660770079172216509622481527302996949162963005704700492517135709241122106134818942813937407221999173792307075575981829352225448946384808227462516479414401300435033896651950865271518471156501887670, 10078670665196003498939913614293154636498704919425436451658616574892145837963206552003620530413131017572295546800910557833093265346618734265862770387825819771608052061514235030272988329322327000124646153905216167043260561754653893047080848786025998767243240213711187565745931784753246023920], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 2, 1], [2, 81, 1, 1], [2, 108, 1, 1], [2, 324, 1, 1], [3, 2, 1, 1], [3, 24, 2, 1], [3, 48, 1, 1], [3, 72, 1, 1], [3, 144, 1, 2], [3, 432, 2, 1], [4, 486, 1, 1], [4, 1458, 1, 1], [6, 162, 1, 1], [6, 216, 1, 1], [6, 216, 2, 1], [6, 648, 1, 1], [6, 648, 2, 3], [6, 1296, 1, 3], [6, 1296, 2, 1], [8, 486, 2, 1], [8, 1458, 1, 2], [9, 864, 1, 1], [12, 972, 1, 1], [24, 972, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3:\\SOPlus(4,2).S_4', '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': 72, 'autcentquo_group': '46656.jt', 'autcentquo_hash': 7932973419676711738, 'autcentquo_nilpotent': False, 'autcentquo_order': 46656, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^3:\\SOPlus(4,2).S_4', 'cc_stats': [[1, 1, 1], [2, 27, 2], [2, 81, 1], [2, 108, 1], [2, 324, 1], [3, 2, 1], [3, 24, 2], [3, 48, 1], [3, 72, 1], [3, 144, 2], [3, 432, 2], [4, 486, 1], [4, 1458, 1], [6, 162, 1], [6, 216, 3], [6, 648, 7], [6, 1296, 5], [8, 486, 2], [8, 1458, 2], [9, 864, 1], [12, 972, 1], [24, 972, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '23328.du', 'commutator_count': 1, 'commutator_label': '5832.ea', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 99, 'cyclic': False, 'derived_length': 6, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 81, 1, 1], [2, 108, 1, 1], [2, 324, 1, 1], [3, 2, 1, 1], [3, 24, 1, 2], [3, 48, 1, 1], [3, 72, 1, 1], [3, 144, 1, 2], [3, 432, 1, 2], [4, 486, 1, 1], [4, 1458, 1, 1], [6, 162, 1, 1], [6, 216, 1, 3], [6, 648, 1, 7], [6, 1296, 1, 5], [8, 486, 2, 1], [8, 1458, 2, 1], [9, 864, 1, 1], [12, 972, 1, 1], [24, 972, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1728, 'exponent': 72, 'exponents_of_order': [6, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[18, 1, 2], [36, 0, 2], [36, 1, 1], [54, 1, 2], [72, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '7776.bh', 'hash': 6694629684818786260, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 72, 'inner_gen_orders': [8, 6], 'inner_gens': [[822419622445341580929239305, 2516890069711663604239518723], [1274400983871152903499874556, 8356766692473337207873663173]], 'inner_hash': 6694629684818786260, 'inner_nilpotent': False, 'inner_order': 23328, 'inner_split': True, 'inner_tex': 'C_3^3:S_3^2.S_4', 'inner_used': [1, 2], 'irrC_degree': 18, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 4], [2, 6], [3, 4], [4, 2], [8, 8], [16, 6], [18, 2], [32, 1], [36, 3], [48, 2], [54, 2], [72, 1]], 'label': '23328.du', 'linC_count': 2, 'linC_degree': 18, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 2, 'linQ_dim': 18, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C3^3:S3^2.S4', 'ngens': 2, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 31, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 41, 'number_divisions': 38, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 625, 'number_subgroup_classes': 953, 'number_subgroups': 137353, 'old_label': None, 'order': 23328, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 567], [3, 1322], [4, 1944], [6, 11826], [8, 3888], [9, 864], [12, 972], [24, 1944]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[9629980800472657548117895085, 1258402146632039681125453584]], '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': 9, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [3, 4], [4, 4], [8, 8], [16, 6], [18, 2], [32, 1], [36, 1], [48, 2], [54, 2], [72, 2]], 'representations': {'PC': {'code': '1045083736268556626812304653039508791097999859400754713314403644908252782167013454461940423225757228398751968658508025356151037155041542547071668467389356710451253772226716586582387653380773253183660765608879318948127', 'gens': [1, 2, 4, 5, 7, 8, 9, 10, 11], 'pres': [11, -2, -2, -3, -2, 2, -2, -3, 3, -3, 3, 3, 28776, 353453, 56, 9374, 421875, 377534, 246997, 212, 904204, 513165, 246371, 367, 158, 1428773, 609856, 346131, 118278, 177425, 181132, 143567, 974, 369, 33799, 71826, 154205, 136616, 755, 1118, 1129, 171080, 85555, 1330569, 522740, 356431, 134682, 75293, 11944, 1406, 1672714, 888645, 182984, 104587, 17478, 43625, 10966]}, 'Perm': {'d': 27, 'gens': [8356766692473337207873663173, 822419622445341580929239305]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3^2.S_4', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
