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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1944.3612', 'ambient_counter': 3612, 'ambient_order': 1944, 'ambient_tex': 'C_3^3:(C_3\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 27, 'characteristic': False, 'core_order': 81, 'counter': 20, 'cyclic': False, 'direct': None, 'hall': 3, 'label': '1944.3612.8.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.a1', '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': 8, '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': '243.62', 'subgroup_hash': 62, 'subgroup_order': 243, 'subgroup_tex': 'C_3^2\\times \\He_3', 'supersolvable': True, 'sylow': 3}
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gps_subgroup_data • Show schema
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{'ambient': '1944.3612', 'aut_centralizer_order': 27, 'aut_label': '8.a1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '11664.hw', 'aut_weyl_index': 108, 'centralizer': '72.h1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1', '4.a1'], 'contains': ['24.a1', '24.b1', '24.c1', '24.d1', '24.e1', '24.f1', '24.g1', '24.h1', '24.i1'], 'core': '24.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [1523, -1, 1112, -1], 'generators': [2, 6, 18, 648], 'label': '1944.3612.8.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.a1', 'old_label': '8.a1', 'projective_image': '1944.3612', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.a1', 'subgroup_fusion': None, 'weyl_group': '18.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '81.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 6, 'aut_exponent': 72, 'aut_gen_orders': [2, 6, 12, 4, 12, 12, 3, 3, 3, 3, 3, 3], 'aut_gens': [[40831, 96397, 80926, 40825, 5941], [40831, 78898, 80926, 40825, 5941], [96397, 61399, 24607, 10942, 6049], [43918, 61399, 41041, 94375, 5941], [61399, 26419, 80926, 40825, 5941], [40939, 96505, 45718, 94375, 5941], [41047, 96505, 41041, 24499, 5941], [40831, 96397, 28429, 43912, 5941], [40831, 96397, 98191, 8914, 5941], [40831, 96397, 98191, 5953, 5941], [40831, 96397, 10930, 26521, 5941], [40831, 96397, 7969, 43804, 5941], [40831, 96397, 42949, 61519, 5941]], 'aut_group': '15116544.lw', 'aut_hash': 2381615621057333688, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 15116544, 'aut_permdeg': 81, 'aut_perms': [90741343241864240220423886686755512566211314393768130638862983616392276903826900609322736102852702738176080189406026, 12224039273171775780865360180656048301319934635678825424790240646525954930592288362030744644039340782699226211597122, 341331181151988697010827124714939447240399610411525694898653811161375848141635113496959477687103657155086642669292900, 83484099268931229354441523075260474297624236377374884088250470974159287698389565469403075151545058318676010450277280, 993383419084969510328867400462007341103952608852298373045281883813123941085657725799292627704965511554093407165742384536, 3218938675203839089292264528254612740273157596997170886316426222426839375765351463712624477320463333671787801499256216191, 578413259912832901573184403138846183539201721630432410514928757113266739709339259656317063932408630342414285272353911, 1521306258310128493580537588823785100633029099363607690618208202466279661782099604623691619231789780762716560279313066464, 434343710029836884363635122254347900645015629734241721435884632255586733353399916675211917551529129034513995213144017000, 433999335735596737840503477066784970256255047790984095244289246758914374121103614332637994873476251607412481344741110760, 290859691956620219904074651963991104896859318712405666164435733526735359709634672226805158599455985579028119854289285048, 434027934894047587880224493289394564713722542080752426774098756141367933780035318020668208130185229773628215508795571690], 'aut_phi_ratio': 93312.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 1, 24, 1], [3, 3, 72, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^6.(C_3^2:\\GL(2,3)\\times \\GL(2,3))', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 72, 'autcent_group': None, 'autcent_hash': 5874786550790147276, 'autcent_nilpotent': False, 'autcent_order': 314928, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_3^4.C_3^4.Q_8.S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '48.29', 'autcentquo_hash': 29, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\GL(2,3)', 'cc_stats': [[1, 1, 1], [3, 1, 26], [3, 3, 72]], 'center_label': '27.5', 'center_order': 27, 'central_product': True, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 62, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['27.3', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 13], [3, 3, 2, 36]], 'element_repr_type': 'GLZN', 'elementary': 3, 'eulerian_function': 130, 'exponent': 3, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '81.15', 'hash': 62, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [1, 1, 3, 3, 1], 'inner_gens': [[40831, 96397, 80926, 40825, 5941], [40831, 96397, 80926, 40825, 5941], [40831, 96397, 80926, 41041, 5941], [40831, 96397, 80710, 40825, 5941], [40831, 96397, 80926, 40825, 5941]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': False, 'inner_tex': 'C_3^2', 'inner_used': [3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 81], [3, 18]], 'label': '243.62', '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*He3', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 99, 'number_divisions': 50, 'number_normal_subgroups': 234, 'number_subgroup_autclasses': 15, 'number_subgroup_classes': 450, 'number_subgroups': 882, 'old_label': None, 'order': 243, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 242]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 72, 'outer_gen_orders': [8, 8, 6], 'outer_gen_pows': [5833, 5833, 24607], 'outer_gens': [[79006, 61507, 80920, 10828, 5941], [96397, 40831, 77002, 80806, 6049], [26311, 78898, 78055, 7012, 5941]], 'outer_group': None, 'outer_hash': 4345039026778699474, 'outer_nilpotent': False, 'outer_order': 1679616, 'outer_permdeg': 45, 'outer_perms': [13754532269951591502539443789292190143512301279954511167, 40263807807184829051873451995385646081346390305766043750, 91541359313162568792597078106269773575721163887177023741], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_3^6.Q_8^2.S_3^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 40], [6, 9]], 'representations': {'PC': {'code': 34816, 'gens': [1, 2, 3, 4, 5], 'pres': [5, -3, 3, 3, 3, -3, 433]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [24505, 5839, 40831, 5941, 96397]}, 'Perm': {'d': 15, 'gens': [19208327187, 114133103524, 206450778484, 280263841440, 280263841200]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3, 3], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2\\times \\He_3', 'transitive_degree': 81, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [6, 12, 6, 6, 6], 'aut_gens': [[1, 6, 18, 54, 324], [11, 660, 840, 54, 1620], [1289, 654, 1068, 270, 1890], [1589, 1302, 1368, 1242, 1458], [1405, 1308, 138, 270, 540], [443, 876, 1110, 54, 432]], 'aut_group': None, 'aut_hash': 5742461704192487224, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1259712, 'aut_permdeg': 168, 'aut_perms': [140893284666465992081439062528457917238398397625592550612859509683579789611263389433313090533556127398496119461248423177471620108163148509993257715905014795029767161537624487454158409698446732885706146261645264085552534393302132501556258226359087817943134187749723660390967052635776580896046329447298424, 71387769141066723981726075021926967368398716050235097378482207824524133222436624913635278867030667236037245343886388778067355859186642291475584534881199496105075677328786281042942501719976806105989093850984165084923831559160877190180853500037475351205295587055800450632807818713792237101692148459848370, 72167474260809314003731089185131102248526174500994611432287404763119778969185502770667539507339558575628732489256864157425625611461822482461095810642070646801395565500971647949296613587284678553074918896356203158697997806065368483644182215162858829185535990628600664568300699735291769504769167398986473, 376833348755648505879483717702874368159727892196402127888894396118022242759863312404184757568529324222275357604106025378912365005796781044331482577230145740346766657736250259165917490001992863121614301696754809952710818983794846451694132577615435824878834350621446108804946766550060719637822403337125, 140197320457540781884805901315360456168741586412095615736860157543705303945466835012380994683668382644734146801433147911744174336119903616233323852915086170549037996018230321699665501108909245087621586116018981863540597847126576521067581010228653693776452221807929349318019487581208014522274983747335063], 'aut_phi_ratio': 1944.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 162, 1, 1], [3, 2, 1, 1], [3, 2, 3, 1], [3, 3, 2, 1], [3, 6, 2, 1], [3, 6, 3, 1], [3, 6, 6, 1], [3, 8, 9, 1], [3, 24, 6, 2], [3, 24, 12, 1], [4, 162, 1, 1], [6, 6, 1, 1], [6, 6, 3, 1], [6, 9, 2, 1], [6, 18, 2, 1], [6, 18, 3, 1], [6, 18, 6, 1], [6, 162, 2, 1], [12, 162, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_6^2.S_3^3.C_2', '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': 36, 'autcentquo_group': None, 'autcentquo_hash': 5742461704192487224, 'autcentquo_nilpotent': False, 'autcentquo_order': 1259712, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_6^2.C_3.D_6^2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 162, 1], [3, 2, 4], [3, 3, 2], [3, 6, 11], [3, 8, 9], [3, 24, 24], [4, 162, 1], [6, 6, 4], [6, 9, 2], [6, 18, 11], [6, 162, 2], [12, 162, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1944.3612', 'commutator_count': 1, 'commutator_label': '324.171', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3612, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 162, 1, 1], [3, 2, 1, 4], [3, 3, 2, 1], [3, 6, 1, 3], [3, 6, 2, 4], [3, 8, 1, 9], [3, 24, 1, 6], [3, 24, 2, 9], [4, 162, 1, 1], [6, 6, 1, 4], [6, 9, 2, 1], [6, 18, 1, 3], [6, 18, 2, 4], [6, 162, 2, 1], [12, 162, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5896800, 'exponent': 12, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '648.737', 'hash': 3612, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 3, 3, 6, 6], 'inner_gens': [[1, 120, 252, 1242, 1620], [229, 6, 18, 54, 324], [145, 6, 18, 1188, 486], [1081, 6, 1152, 54, 324], [649, 6, 180, 54, 324]], 'inner_hash': 3612, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': False, 'inner_tex': 'C_3^3:(C_3\\times S_4)', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 39], [3, 6], [6, 21], [18, 3]], 'label': '1944.3612', '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': 'C3^3:(C3*S4)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 75, 'number_divisions': 54, 'number_normal_subgroups': 96, 'number_subgroup_autclasses': 239, 'number_subgroup_classes': 1005, 'number_subgroups': 20020, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 165], [3, 728], [4, 162], [6, 564], [12, 324]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [3, 6, 3, 3, 3, 2, 2], 'outer_gen_pows': [3, 0, 0, 3, 0, 0, 0], 'outer_gens': [[1, 12, 1440, 1242, 1620], [1, 12, 144, 1242, 324], [1, 1302, 1314, 54, 324], [1, 12, 684, 1242, 1620], [1, 114, 882, 54, 432], [5, 228, 36, 1026, 324], [1, 6, 150, 1026, 324]], 'outer_group': '648.555', 'outer_hash': 555, 'outer_nilpotent': False, 'outer_order': 648, 'outer_permdeg': 12, 'outer_perms': [120114720, 121214282, 787948, 2219904, 328544040, 9441506, 39937944], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3.S_3^3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 15], [3, 2], [4, 13], [6, 15], [12, 4], [18, 3]], 'representations': {'PC': {'code': 615342338804487498494798894649800094238000819478679925, 'gens': [1, 3, 4, 5, 7], 'pres': [8, 2, 3, 3, 3, 2, 3, 2, 3, 16, 2882, 1378, 8067, 3755, 49684, 2668, 116, 10373, 90726, 1542, 166, 82951]}, 'Perm': {'d': 16, 'gens': [180587244289, 16313, 93448857600, 79833600, 43597532, 98499, 1313901388800, 2789705318400]}}, 'schur_multiplier': [3, 3, 3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:(C_3\\times S_4)', 'transitive_degree': 108, 'wreath_data': None, 'wreath_product': False}
