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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '768.1085882', 'ambient_counter': 1085882, 'ambient_order': 768, 'ambient_tex': 'C_2^3.\\GL(2,\\mathbb{Z}/4)', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': True, 'core_order': 384, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '768.1085882.2.b1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '384.5721', 'subgroup_hash': 5721, 'subgroup_order': 384, 'subgroup_tex': 'C_2^2.\\GL(2,\\mathbb{Z}/4)', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '768.1085882', 'aut_centralizer_order': 2, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1536.408616173', 'aut_weyl_index': 2, 'centralizer': '384.a1.a1', 'complements': ['384.e1.a1', '384.e1.b1', '384.i1.a1', '384.i1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.b1.a1', '4.f1.a1', '6.f1.a1', '8.g1.a1'], 'core': '2.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3558, 2430, 7611, 4617, 4687, 2307, 5556, 2683], 'generators': [41464805, 98155440, 23, 136095240, 18558000, 172200, 3, 16], 'label': '768.1085882.2.b1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.b1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.b1.a1', 'projective_image': '384.5721', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.b1.a1', 'subgroup_fusion': None, 'weyl_group': '384.5721'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', '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, 2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[44392442, 41141520, 90882840, 94525320, 139725360, 3, 7, 16], [44392442, 41141520, 90882840, 94525320, 139725360, 4, 16, 7], [44392441, 41141520, 90882840, 94525320, 139725360, 3, 16, 23], [83997482, 41141520, 125199480, 94525320, 139725360, 3, 7, 16], [159749282, 207290280, 90882840, 94525320, 139725360, 3, 7, 16], [248250362, 41141520, 298746840, 295108680, 139725360, 3, 7, 16], [44392461, 41141520, 90882840, 94525320, 139725360, 11, 7, 16], [44392442, 41141520, 90882840, 94525320, 139725360, 12, 7, 16], [263138402, 251522640, 90882840, 94525320, 139725360, 3, 7, 16], [201163082, 41141520, 298746840, 94525320, 139725360, 3, 7, 16], [263138402, 41141520, 90882840, 94525320, 139725360, 3, 7, 16]], 'aut_group': '1536.408616173', 'aut_hash': 9144891369713934392, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 96, 'aut_perms': [10114147405445231965811398858795835412714885021341057850725452195428352010274124144003054516660542026210568093310318760884102424261531477709773061062, 129920107665811769699960724136072851041798037318011211307134710418784770924529412024853047294294381665030818980024239609499603753708874906641816689870, 646235485793398156474032713703873118862380772606677877330308357195812908376720183262446065779402104426546990101447228718772813677410009808490370600761, 660316346369876745252901582978479722214542440999382041557224235152049859769424188445006327719134122604875293628865270338516942690562981667236173091971, 526054849460389558145327692828046923028893427786136394960147078954863263167815357961020982464655882974205167305849480179227862326506171138805939426963, 857553670243096110183910847353619251586205160013559393153248586925220471722527158756847171801201475075608693847957726742646889814861512919621074675115, 8483583329531293217006571879024388282889953643183810007131555589499690005190515505648587334201974871620478132048795739409508139370290304672438745620, 702650251719292816835073119359884208557031423940651631921212763121084427958839111882257324600857887191851805220409257308411145092022563032107447130840, 847451559405738866823588195642241371319181568042822032611883421928626228577391684960483762810690740474581711058793898608699422825961589854717656516194, 703411733044309171954333634153948054766553020496749143265492083519101265857782211997077726172127107657929276604180986719489721969896698237349042600862], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 2, 2, 1], [2, 3, 1, 2], [2, 4, 1, 1], [2, 6, 1, 1], [2, 6, 2, 1], [2, 12, 1, 1], [3, 8, 1, 1], [4, 4, 1, 1], [4, 12, 1, 1], [4, 24, 4, 2], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 2, 2], [12, 16, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\wr C_2^2\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '384.20051', 'autcentquo_hash': 20051, 'autcentquo_nilpotent': False, 'autcentquo_order': 384, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\GL(2,\\mathbb{Z}/4):C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 3, 2], [2, 4, 1], [2, 6, 3], [2, 12, 1], [3, 8, 1], [4, 4, 1], [4, 12, 1], [4, 24, 8], [6, 8, 3], [6, 16, 4], [12, 16, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '192.991', 'commutator_count': 1, 'commutator_label': '48.49', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 5721, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 3, 1, 2], [2, 4, 1, 1], [2, 6, 1, 3], [2, 12, 1, 1], [3, 8, 1, 1], [4, 4, 1, 1], [4, 12, 1, 1], [4, 24, 2, 4], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 1, 2], [6, 16, 2, 1], [12, 16, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 18, 'exponent': 12, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 1]], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '48.48', 'hash': 5721, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 2, 2, 2, 1, 3, 2, 2], 'inner_gens': [[44392442, 207290280, 90882840, 295108680, 139725360, 4, 16, 7], [201163082, 41141520, 298746840, 94525320, 139725360, 3, 7, 16], [44392442, 251522640, 90882840, 94525320, 139725360, 3, 7, 16], [263138402, 41141520, 90882840, 94525320, 139725360, 3, 7, 16], [44392442, 41141520, 90882840, 94525320, 139725360, 3, 7, 16], [44392441, 41141520, 90882840, 94525320, 139725360, 3, 16, 23], [44392461, 41141520, 90882840, 94525320, 139725360, 20, 7, 16], [44392461, 41141520, 90882840, 94525320, 139725360, 11, 7, 16]], 'inner_hash': 991, 'inner_nilpotent': False, 'inner_order': 192, 'inner_split': True, 'inner_tex': 'C_2.\\GL(2,\\mathbb{Z}/4)', 'inner_used': [1, 2, 6, 7], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 10], [3, 8], [4, 3], [6, 2], [12, 1]], 'label': '384.5721', 'linC_count': 24, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 4, 'linQ_dim': 7, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2^2.GL(2,Z/4)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 32, 'number_divisions': 25, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 196, 'number_subgroup_classes': 283, 'number_subgroups': 1700, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 47], [3, 8], [4, 208], [6, 88], [12, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[83997482, 41141520, 125199480, 94525320, 139725360, 3, 7, 16], [248250362, 41141520, 298746840, 295108680, 139725360, 3, 7, 16], [44392442, 41141520, 90882840, 94525320, 139725360, 4, 16, 7]], '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': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [3, 4], [4, 3], [6, 4], [8, 1], [12, 1]], 'representations': {'PC': {'code': 40033817689061328645319281527533982096332083370244, 'gens': [1, 3, 4, 7, 8], 'pres': [8, -2, -2, -2, 2, -2, -3, -2, 2, 16, 1250, 2947, 907, 91, 3204, 116, 3077, 16134, 2046, 710, 550, 799, 1191, 239]}, 'Perm': {'d': 12, 'gens': [44392442, 41141520, 90882840, 94525320, 139725360, 3, 7, 16]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2.\\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 24, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 6, 4, 2, 2, 2, 2, 2], 'aut_gens': [[41464805, 90882840, 136095240, 172200, 18558000, 98155440, 3, 7, 16], [41495774, 11290320, 132452760, 172200, 18558000, 98155440, 20, 7, 16], [239637614, 11290320, 295108680, 98023680, 18558000, 98155440, 20, 7, 16], [41495765, 90882840, 298746840, 172200, 18558000, 98155440, 11, 7, 16], [41495774, 90882840, 298746840, 172200, 18558000, 98155440, 20, 7, 16], [41495761, 90882840, 298746840, 172200, 18558000, 98155440, 3, 23, 7], [41464805, 302376360, 132452760, 172200, 83629560, 98155440, 11, 7, 16], [41464805, 94525320, 295108680, 172200, 18558000, 98155440, 3, 7, 16], [182931845, 90882840, 136095240, 18679680, 18558000, 98155440, 3, 7, 16], [247217774, 94525320, 132452760, 172200, 18558000, 98155440, 20, 7, 16], [247217765, 94525320, 132452760, 172200, 18558000, 98155440, 3, 7, 16], [41464805, 90882840, 136095240, 172200, 18558000, 98155440, 4, 23, 16]], 'aut_group': '3072.bhj', 'aut_hash': 6634943677507123677, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3072, 'aut_permdeg': 104, 'aut_perms': [7338415246152560845531686058214253162144392778019283303409034695496086734531821033250218207625173372134857391339976105806414522830137070705660028365052252647394916676, 50968503072943487039312884902663351768952685657429423377760089073296391470986710423238524799325705497796222794199432068945662731050873713411557622839478050815869025, 525682564363103633780448182781112409089041405580061802831345588156452541412997552776050110925513313077151899025700169217374946335464770553342959453646115259090516449, 3087477132428544924220953146429900087505877164140062265044298949924951943720986265931814434029292057208238563254489489274929120080615494213684608095435946431317059645, 8983402821553279844898172718755278395828703776445807271212842053632225469917236021218450519359801004706445286869947541084899944403259156734841880014908598243693422121, 1327014201767702642641943559322748647425938953401041895130455775464300766990947932180726572360990298433010004166316430500746924675127711958646956650426595575029739521, 9020124777240522839610977090719678685414091964610990486214701789936804118149738834140734462651485606424863254538550209214329994413601713687345061016071060977735086916, 401310473191953255330016365353048355622766337684275742493984651108352994608425145773251852360954536717428655977083170540328251861184273234629133368206635959518444630, 7338415246152560844444319901556953633943967782308207089556541893758725432752980001063421400579380115682294226827841515216749305769375475043991477101769203904482916676, 7352835452073557732630556033090335540796794703022759523760970961067976962284086515142963848122480644355979273175783708704761361602085084220606384741428668516115231842, 5844031276530809512013660578870729997456142935250222154386184814921082345489826185588124629300320962398861048441106996579739918299448929921967765501507404936409646536], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 4, 1, 2], [2, 4, 2, 1], [2, 6, 1, 1], [2, 12, 1, 2], [2, 12, 2, 1], [3, 8, 1, 1], [4, 4, 1, 1], [4, 8, 1, 1], [4, 12, 1, 1], [4, 24, 1, 1], [4, 48, 2, 2], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 2, 1], [6, 16, 4, 1], [6, 32, 1, 1], [8, 48, 2, 2], [12, 32, 1, 1], [12, 32, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:D_4\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '768.1088763', 'autcentquo_hash': 1088763, 'autcentquo_nilpotent': False, 'autcentquo_order': 768, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^6:D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 3, 2], [2, 4, 4], [2, 6, 1], [2, 12, 4], [3, 8, 1], [4, 4, 1], [4, 8, 1], [4, 12, 1], [4, 24, 1], [4, 48, 4], [6, 8, 3], [6, 16, 6], [6, 32, 1], [8, 48, 4], [12, 32, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '384.5721', 'commutator_count': 1, 'commutator_label': '96.228', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1085882, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 4, 1, 4], [2, 6, 1, 1], [2, 12, 1, 4], [3, 8, 1, 1], [4, 4, 1, 1], [4, 8, 1, 1], [4, 12, 1, 1], [4, 24, 1, 1], [4, 48, 2, 2], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 2, 3], [6, 32, 1, 1], [8, 48, 2, 2], [12, 32, 1, 1], [12, 32, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 36, 'exponent': 24, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 2]], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '48.48', 'hash': 1085882, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 2, 2, 2, 2, 1, 3, 2, 2], 'inner_gens': [[41464805, 302376360, 132452760, 172200, 83629560, 98155440, 4, 23, 16], [211690205, 90882840, 136095240, 18679680, 18558000, 98155440, 3, 7, 16], [239637605, 90882840, 136095240, 98023680, 18558000, 98155440, 3, 7, 16], [41464805, 94525320, 295108680, 172200, 18558000, 98155440, 3, 7, 16], [247248725, 90882840, 136095240, 172200, 18558000, 98155440, 3, 7, 16], [41464805, 90882840, 136095240, 172200, 18558000, 98155440, 3, 7, 16], [41464802, 90882840, 136095240, 172200, 18558000, 98155440, 3, 16, 23], [41464814, 90882840, 136095240, 172200, 18558000, 98155440, 20, 7, 16], [41464805, 90882840, 136095240, 172200, 18558000, 98155440, 11, 7, 16]], 'inner_hash': 5721, 'inner_nilpotent': False, 'inner_order': 384, 'inner_split': True, 'inner_tex': 'C_2^2.\\GL(2,\\mathbb{Z}/4)', 'inner_used': [1, 2, 7, 8], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 10], [3, 8], [4, 9], [6, 2], [12, 3]], 'label': '768.1085882', 'linC_count': 48, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 8, 'linQ_dim': 7, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.GL(2,Z/4)', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 28, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 40, 'number_divisions': 31, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 457, 'number_subgroup_classes': 634, 'number_subgroups': 5522, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 79], [3, 8], [4, 240], [6, 152], [8, 192], [12, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[41495765, 90882840, 298746840, 172200, 18558000, 98155440, 3, 7, 16], [41464805, 14928480, 136095240, 172200, 18558000, 98155440, 3, 7, 16], [41464805, 90882840, 136095240, 172200, 18558000, 98155440, 4, 23, 16]], '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': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [3, 4], [4, 5], [6, 4], [8, 3], [12, 3]], 'representations': {'PC': {'code': '528783370763308613158415767814894723753941221287757950859620872', 'gens': [1, 3, 4, 7, 8, 9], 'pres': [9, -2, -2, -2, 2, -2, -3, -2, 2, -2, 18, 11774, 5249, 17139, 1020, 3549, 102, 20884, 130, 3461, 1545, 1176, 429, 20743, 2626, 475, 700]}, 'Perm': {'d': 12, 'gens': [41464805, 90882840, 136095240, 172200, 18558000, 98155440, 3, 7, 16]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3.\\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
