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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '5832.kc', 'ambient_counter': 263, 'ambient_order': 5832, 'ambient_tex': 'C_3^4:(C_3\\times D_{12})', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 27, 'counter': 70, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5832.kc.18.j1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '18.j1', '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': 18, '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': '324.166', 'subgroup_hash': 166, 'subgroup_order': 324, 'subgroup_tex': 'C_3^2:S_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5832.kc', 'aut_centralizer_order': None, 'aut_label': '18.j1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1944.c1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.g1', '6.j1'], 'contains': ['36.f1', '36.cl1', '36.cw1', '54.f1', '54.u1', '54.v1', '54.w1', '54.x1'], 'core': '216.a1', 'coset_action_label': None, 'count': 18, 'diagramx': [7387, -1, 8148, -1], 'generators': [109227, 4103606707200, 64076044032000, 207171, 4016512167213, 79833600], 'label': '5832.kc.18.j1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.j1', 'old_label': '18.j1', 'projective_image': '5832.kc', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.j1', 'subgroup_fusion': None, 'weyl_group': '108.39'}
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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': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 3, 6, 4, 4, 3, 4, 4, 3, 2], 'aut_gens': [[1, 6, 18, 108], [79, 12, 90, 108], [73, 6, 18, 108], [49, 12, 198, 108], [85, 84, 24, 108], [37, 48, 30, 108], [49, 6, 18, 108], [13, 72, 66, 216], [5, 78, 96, 108], [7, 6, 24, 108], [73, 6, 90, 108]], 'aut_group': '5184.qr', 'aut_hash': 8627901162325749137, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5184, 'aut_permdeg': 14, 'aut_perms': [56567888771, 25517330398, 34180400329, 69023093998, 50338967290, 81717529188, 25391078837, 81848719572, 2546461, 3237814], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 2, 4, 1], [3, 2, 8, 1], [3, 4, 4, 1], [3, 4, 8, 1], [6, 3, 2, 1], [6, 6, 4, 1], [6, 6, 8, 1], [6, 9, 2, 1], [6, 18, 1, 1], [6, 18, 2, 1], [6, 27, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2:\\GL(2,3)\\times D_6', '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': 24, 'autcentquo_group': '2592.fv', 'autcentquo_hash': 9019849488891309561, 'autcentquo_nilpotent': False, 'autcentquo_order': 2592, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 1], [3, 1, 2], [3, 2, 15], [3, 4, 12], [6, 3, 2], [6, 6, 12], [6, 9, 2], [6, 18, 3], [6, 27, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '108.39', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 166, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 1, 2, 1], [3, 2, 1, 5], [3, 2, 2, 5], [3, 4, 1, 4], [3, 4, 2, 4], [6, 3, 2, 1], [6, 6, 1, 4], [6, 6, 2, 4], [6, 9, 2, 1], [6, 18, 1, 1], [6, 18, 2, 1], [6, 27, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2184, 'exponent': 6, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '324.166', 'hash': 166, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 6, 3], 'inner_gens': [[1, 12, 90, 108], [13, 6, 18, 108], [37, 6, 18, 216], [1, 6, 234, 108]], 'inner_hash': 39, 'inner_nilpotent': False, 'inner_order': 108, 'inner_split': False, 'inner_tex': 'C_3:S_3^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 12], [2, 30], [4, 12]], 'label': '324.166', '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': 'C3^2:S3^2', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 18, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 54, 'number_divisions': 36, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 87, 'number_subgroup_classes': 224, 'number_subgroups': 1000, 'old_label': None, 'order': 324, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 39], [3, 80], [6, 204]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 0, 1, 1, 1], 'outer_gens': [[1, 78, 90, 108], [5, 12, 90, 108], [1, 48, 90, 108], [1, 36, 60, 108], [1, 42, 102, 108]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [598, 719, 373, 639, 86], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 14], [4, 14], [8, 4]], 'representations': {'PC': {'code': 241842432640628585051, 'gens': [1, 3, 4, 6], 'pres': [6, -2, -3, -3, -2, -3, -3, 12, 218, 2163, 69, 2164, 455]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [24505, 40831, 6083, 5941, 96397, 5995]}, 'Perm': {'d': 12, 'gens': [136, 3633840, 325, 43590960, 45360, 435]}}, 'schur_multiplier': [6], '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_3^2:S_3^2', 'transitive_degree': 36, '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': '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': [6, 6, 6, 6, 12, 12], 'aut_gens': [[43153257785482, 379399966894951], [87218450041, 443301690618816], [87262042413, 1111529519667082], [64163226015253, 1510370202024932], [1800761013710016, 1154850867135253], [4016432219598, 1067068591272932], [1800761050105607, 1510370202005631]], 'aut_group': None, 'aut_hash': 375384708104923416, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 46656, 'aut_permdeg': 186, 'aut_perms': [2234400006492680047188868741477499249179230425357868436944652339474197124980789263260452836353621971239289712362210067326235441460655032383696076244699107863259019248073980038992861293687995768564876327659824421004214777758405370496056294393349599854952508360956419927216728362659460437549316998139400411949847961236025821361337872382983348157, 4970577951685414795370386809147012320387733581690501772888881897464337564334430905345409362772598663598252379937428599607913802561406500593886889440247023910090650109581631710256263919957233010884228743367005923605859575443776314913879177073591187507587576479490672087286902009490781882123952042677175376813225204199983144728633359163874636637, 4942912221735138663134947329774231150269821477125303647075625128060372529588741991603327745148035615657528200113045465245667852044875709435682445588609484451646964560508349043944852585133911194576689180696238043142978511247283225488107295622402319867630737154954274021327109420296072516255506805487562614061293691621454637562042869716956532489, 381315989509859615860727305507770727937955533657256418344882184209067426379631494342831915541547689425369164183613169837975475011306397711768542118287409629456962247068199771356629676110480953071963941313148810976946313485069413369234609433112730370454400881210773393757780011255546021654580069693169692098513582578253524538182584636094414090, 214454615453071633818921304836710737190162762069976489280021953462562983315395635090350335296681915749463077108413849659196451914599914187387443221043798733597234423615988824870985851189140057738968755457960449646558677199368589409768542187920935833828126878820366191121126002587587149234409210538824177529053204414381255348754739214656319948, 6362902578607376271883435702443363351818938177014311856456608103009211695654428787416791566281767078464998570348318007552624105075154293623206046476441312559570231532317381196525153766420684134541960678991650556604929007205402540599222297550331041220191286935265288349305906067369648800585545092878184460017254245849506432411974379303267531541], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 18, 1, 1], [2, 162, 1, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 3], [3, 4, 2, 1], [3, 6, 1, 1], [3, 6, 2, 2], [3, 8, 1, 3], [3, 8, 2, 2], [3, 12, 1, 1], [3, 12, 2, 4], [3, 12, 4, 1], [3, 24, 1, 1], [3, 24, 2, 5], [3, 24, 4, 2], [4, 162, 1, 1], [6, 18, 1, 2], [6, 18, 2, 1], [6, 27, 2, 1], [6, 36, 1, 2], [6, 36, 2, 1], [6, 54, 1, 1], [6, 54, 2, 4], [6, 54, 4, 1], [6, 108, 1, 1], [6, 108, 2, 3], [6, 108, 4, 1], [6, 162, 2, 1], [6, 324, 1, 1], [6, 324, 2, 1], [12, 162, 2, 2], [12, 162, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^5.C_6.C_2^5', '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': 12, 'autcentquo_group': None, 'autcentquo_hash': 375384708104923416, 'autcentquo_nilpotent': False, 'autcentquo_order': 46656, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^5.C_6.C_2^5', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 18, 1], [2, 162, 1], [3, 2, 2], [3, 3, 2], [3, 4, 5], [3, 6, 5], [3, 8, 7], [3, 12, 13], [3, 24, 19], [4, 162, 1], [6, 18, 4], [6, 27, 2], [6, 36, 4], [6, 54, 13], [6, 108, 11], [6, 162, 2], [6, 324, 3], [12, 162, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '5832.kc', 'commutator_count': 1, 'commutator_label': '486.257', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 263, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 18, 1, 1], [2, 162, 1, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 3], [3, 4, 2, 1], [3, 6, 1, 1], [3, 6, 2, 2], [3, 8, 1, 5], [3, 8, 2, 1], [3, 12, 1, 1], [3, 12, 2, 6], [3, 24, 1, 3], [3, 24, 2, 8], [4, 162, 1, 1], [6, 18, 1, 2], [6, 18, 2, 1], [6, 27, 2, 1], [6, 36, 1, 2], [6, 36, 2, 1], [6, 54, 1, 1], [6, 54, 2, 6], [6, 108, 1, 1], [6, 108, 2, 5], [6, 162, 2, 1], [6, 324, 1, 1], [6, 324, 2, 1], [12, 162, 2, 2], [12, 162, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 96, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 0, 2], [24, 1, 2]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '1944.3912', 'hash': 2627231563110374959, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 12], 'inner_gens': [[43153257785482, 1154850903500807], [4016512167213, 379399966894951]], 'inner_hash': 2627231563110374959, 'inner_nilpotent': False, 'inner_order': 5832, 'inner_split': True, 'inner_tex': 'C_3^4:(C_3\\times D_{12})', 'inner_used': [1, 2], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 12], [2, 27], [4, 30], [6, 4], [8, 21], [12, 6], [24, 5]], 'label': '5832.kc', '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': True, 'name': 'C3^4:(C3*D12)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 58, 'number_characteristic_subgroups': 43, 'number_conjugacy_classes': 105, 'number_divisions': 65, 'number_normal_subgroups': 43, 'number_subgroup_autclasses': 1198, 'number_subgroup_classes': 1768, 'number_subgroups': 41688, 'old_label': None, 'order': 5832, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 189], [3, 728], [4, 162], [6, 3456], [12, 1296]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[43153258014841, 379399966881216], [43153258041132, 379399966959231], [43153294073482, 379400003182951]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 9], [4, 15], [6, 2], [8, 18], [12, 3], [16, 8], [24, 5], [48, 1]], 'representations': {'PC': {'code': '94126295119084822981153933729960367250918977405941536964907292618955219363699359', 'gens': [1, 3, 6, 7, 8, 9], 'pres': [9, -2, -3, -2, -2, -3, -3, -3, -3, 3, 18, 147584, 74, 118803, 102, 2164, 311045, 101102, 24647, 17852, 27222, 18168, 18177, 373255, 62233, 7810, 43766, 26279]}, 'Perm': {'d': 18, 'gens': [43153257785482, 379399966894951]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:(C_3\\times D_{12})', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
