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gps_subgroup_search • Show schema
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{'Agroup': False, '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': 9, 'counter': 57, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5832.kc.12.z1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.z1', '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': 12, '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': '486.229', 'subgroup_hash': 229, 'subgroup_order': 486, 'subgroup_tex': 'C_3^4:C_6', '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': '12.z1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1944.f1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['4.c1'], 'contains': ['24.q1', '36.co1', '36.cp1', '36.cq1', '36.cr1', '36.di1', '36.dn1', '36.dq1'], 'core': '648.b1', 'coset_action_label': None, 'count': 24, 'diagramx': [6181, -1, 6958, -1], 'generators': [64076044032000, 80158173, 43805266, 207171, 1799453415513600, 4016512167213], 'label': '5832.kc.12.z1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '12.z1', 'old_label': '12.z1', 'projective_image': '5832.kc', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.z1', 'subgroup_fusion': None, 'weyl_group': '162.40'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [6, 6, 6, 6, 18], 'aut_gens': [[40831, 99463, 10726, 61399, 5941], [40831, 100339, 42004, 61291, 5941], [75829, 68323, 44857, 9028, 5941], [75829, 49786, 63205, 8812, 6049], [40831, 101335, 44749, 8812, 6049], [75829, 51754, 63421, 61507, 5941]], 'aut_group': None, 'aut_hash': 4701574152767778616, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 34992, 'aut_permdeg': 45, 'aut_perms': [60617932662099917035444080508485812635910028902872269368, 7432574398498026302247290679363960372386075405961097222, 54083438195318011699566344731478357139329610758528580941, 9030598956651128691313349637130922792974406078064638416, 91552285961632838669841378988566493481424746358216931903], 'aut_phi_ratio': 216.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 2, 3, 1], [3, 2, 6, 1], [3, 3, 6, 1], [3, 6, 3, 1], [3, 6, 6, 2], [3, 6, 18, 1], [6, 27, 2, 1], [6, 27, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^4:C_3.D_6^2', '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': 18, 'autcentquo_group': '5832.et', 'autcentquo_hash': 7243380119719690966, 'autcentquo_nilpotent': False, 'autcentquo_order': 5832, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^3.S_3^3', 'cc_stats': [[1, 1, 1], [2, 27, 1], [3, 1, 2], [3, 2, 12], [3, 3, 6], [3, 6, 33], [6, 27, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '162.40', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 229, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['162.40', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [3, 1, 2, 1], [3, 2, 1, 4], [3, 2, 2, 4], [3, 3, 2, 3], [3, 6, 1, 3], [3, 6, 2, 15], [6, 27, 2, 4]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 1456, 'exponent': 6, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '162.52', 'hash': 229, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 6, 3, 3, 3], 'inner_gens': [[40831, 99463, 10726, 61399, 5941], [40831, 99463, 94483, 8920, 6049], [40831, 16768, 10726, 61399, 5941], [40831, 102217, 10726, 61399, 5941], [40831, 99355, 10726, 61399, 5941]], 'inner_hash': 40, 'inner_nilpotent': False, 'inner_order': 162, 'inner_split': False, 'inner_tex': 'C_3^3:C_6', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 18], [2, 36], [6, 9]], 'label': '486.229', '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:C6', 'ngens': 6, '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': 14, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 63, 'number_divisions': 36, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 95, 'number_subgroup_classes': 333, 'number_subgroups': 1554, 'old_label': None, 'order': 486, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 27], [3, 242], [6, 216]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 3, 3, 3], 'outer_gen_pows': [6061, 5833, 5833, 5833, 5833, 5833], 'outer_gens': [[75829, 99463, 10942, 61399, 5941], [75829, 99349, 7855, 8920, 6049], [75829, 99349, 10942, 61399, 6049], [40831, 99463, 10726, 61507, 5941], [40831, 99463, 7963, 61399, 5941], [40831, 64351, 10726, 61399, 5941]], 'outer_group': '216.162', 'outer_hash': 162, 'outer_nilpotent': False, 'outer_order': 216, 'outer_permdeg': 9, 'outer_perms': [178680, 1, 138240, 45600, 3, 80880], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3^3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 12], [4, 16], [6, 3], [12, 3]], 'representations': {'PC': {'code': 7495563909149884591, 'gens': [1, 2, 4, 5, 6], 'pres': [6, -3, -2, -3, -3, -3, -3, 31, 2889, 1383, 1090, 3899]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [24505, 5839, 40831, 99235, 5941, 96397]}, 'Perm': {'d': 15, 'gens': [479369656, 107417283120, 325, 25438618800, 200237910195, 292720826160]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:C_6', 'transitive_degree': 54, '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}
