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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.9073', 'ambient_counter': 9073, 'ambient_order': 1344, 'ambient_tex': '(C_2\\times C_{12}):D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 4, 'counter': 187, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.9073.84.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '84.f1', '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': 84, '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': '16.11', 'subgroup_hash': 11, 'subgroup_order': 16, 'subgroup_tex': 'C_2\\times D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.9073', 'aut_centralizer_order': None, 'aut_label': '84.f1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '168.b1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['12.i1', '28.h1', '42.a1', '42.f1', '42.g1'], 'contains': ['168.e1', '168.f1', '168.g1', '168.s1'], 'core': '336.a1', 'coset_action_label': None, 'count': 84, 'diagramx': None, 'generators': [899, 234, 16], 'label': '1344.9073.84.f1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.f1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '21.a1', 'old_label': '84.f1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '84.f1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2], 'aut_gens': [[126, 55, 289, 288], [127, 55, 289, 288], [126, 265, 1, 288], [54, 127, 289, 288], [414, 55, 289, 288], [127, 54, 289, 288], [414, 265, 289, 288]], 'aut_group': '64.138', 'aut_hash': 138, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 8, 'aut_perms': [2309, 526, 5329, 3043, 12316, 18498], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [4, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [4, 2, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [4, 2, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 21, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 1], 'inner_gens': [[126, 265, 289, 288], [414, 55, 289, 288], [126, 55, 289, 288], [126, 55, 289, 288]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.11', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 8, 'linQ_dim': 3, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 10, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 27, 'number_subgroups': 35, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 11], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 151, 0], 'outer_gens': [[54, 127, 289, 288], [127, 55, 289, 288], [415, 54, 1, 288], [127, 54, 289, 288]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2]], 'representations': {'PC': {'code': 8772, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 78, 34]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16322, 16432, 3198]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8912, 8156, 13286, 14044]}, 'Perm': {'d': 6, 'gens': [126, 55, 289, 288]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times D_4', 'transitive_degree': 8, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [12, 12, 28, 6, 4, 12, 4, 14], 'aut_gens': [[1, 2, 8, 32], [597, 246, 24, 352], [405, 678, 24, 804], [1277, 462, 28, 948], [769, 470, 28, 164], [389, 1138, 12, 1328], [1157, 1126, 8, 560], [1173, 694, 28, 1328], [597, 454, 28, 928]], 'aut_group': None, 'aut_hash': 2746774753782569610, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 258048, 'aut_permdeg': 104, 'aut_perms': [206363952208996617832145610495920186233998449317348486076896518328530258929556428595554120168025297792491880800445429410346997189300040339746178963975595934807594448, 9717805694964764455783166804333390692756497445032400397506516870350725093514747803383847218882674756501541215483109757474834369715054513513719032657199606243596797220, 4571689343315135498260944751591941961615072755698168886174854870662558743662230027345213257092240338237217396961267814502355741909466079008896826966220856492685362812, 9160610321312952144033759090603863798146477333903174824431066011505811229366629087316836213500270672528023759399713100252354161306804212852381165438165076490371618197, 9199757660824608856898815929851993355788922878051023863891488265589084801217842946847816753414147796538527596647846801681723267509854213437933627234332418640057034913, 9994909748867825099241798609691224868171494973233946764022224669214501808213219610111755192520817199655158537717649046034178213281070293126979441260991052131300753372, 5022602785145048892495631541334645718751973673529714243364334708293644235991015616012481580015450252877264833677078016880583892978218297945127154981841836620594103474, 4542413937994456673064476826743966934149953237263640742726294820792152421588435451460443819493401465630789282201932521581055158891188268603047392239469617504041382550], 'aut_phi_ratio': 672.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 28, 2, 1], [2, 42, 4, 1], [3, 2, 1, 1], [4, 1, 4, 1], [4, 2, 2, 1], [4, 12, 4, 1], [4, 28, 2, 1], [4, 42, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 2, 1], [6, 28, 4, 1], [7, 2, 3, 1], [12, 2, 4, 1], [12, 4, 2, 1], [12, 28, 4, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 4, 6, 1], [21, 4, 3, 1], [28, 2, 12, 1], [28, 4, 6, 1], [28, 12, 24, 1], [42, 4, 3, 1], [42, 4, 6, 1], [42, 4, 12, 1], [84, 4, 12, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '1008.906', 'autcentquo_hash': 906, 'autcentquo_nilpotent': False, 'autcentquo_order': 1008, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 28, 2], [2, 42, 4], [3, 2, 1], [4, 1, 4], [4, 2, 2], [4, 12, 4], [4, 28, 2], [4, 42, 4], [6, 2, 3], [6, 4, 2], [6, 28, 4], [7, 2, 3], [12, 2, 4], [12, 4, 2], [12, 28, 4], [14, 2, 9], [14, 4, 6], [21, 4, 3], [28, 2, 12], [28, 4, 6], [28, 12, 24], [42, 4, 21], [84, 4, 24]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '168.50', 'commutator_count': 2, 'commutator_label': '84.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 9073, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 28, 1, 2], [2, 42, 1, 4], [3, 2, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 12, 1, 4], [4, 28, 1, 2], [4, 42, 2, 2], [6, 2, 1, 3], [6, 4, 1, 2], [6, 28, 2, 2], [7, 2, 3, 1], [12, 2, 2, 2], [12, 4, 1, 2], [12, 28, 2, 2], [14, 2, 3, 3], [14, 4, 3, 2], [21, 4, 3, 1], [28, 2, 6, 2], [28, 4, 3, 2], [28, 12, 6, 4], [42, 4, 3, 3], [42, 4, 6, 2], [84, 4, 6, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3734640, 'exponent': 84, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '336.219', 'hash': 9073, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 2, 1, 42], 'inner_gens': [[1, 6, 8, 432], [5, 2, 8, 944], [1, 2, 8, 32], [977, 466, 8, 32]], 'inner_hash': 50, 'inner_nilpotent': False, 'inner_order': 168, 'inner_split': False, 'inner_tex': 'S_3\\times D_{14}', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 76], [4, 64]], 'label': '1344.9073', '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': '(C2*C12):D28', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 32, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 156, 'number_divisions': 62, 'number_normal_subgroups': 164, 'number_subgroup_autclasses': 264, 'number_subgroup_classes': 660, 'number_subgroups': 5684, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 231], [3, 2], [4, 280], [6, 126], [7, 6], [12, 128], [14, 42], [21, 12], [28, 336], [42, 84], [84, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [12, 6, 6, 4, 12], 'outer_gen_pows': [0, 0, 0, 832, 0], 'outer_gens': [[1169, 694, 28, 612], [209, 18, 12, 164], [505, 254, 12, 820], [873, 254, 24, 36], [1277, 14, 28, 992]], 'outer_group': '1536.519620', 'outer_hash': 7012476704417447008, 'outer_nilpotent': True, 'outer_order': 1536, 'outer_permdeg': 192, 'outer_perms': [139137412012702207733471497143320589859808489081746760974647030154114926081066870854820604437940451949262332938624465926560715248984655275800626266878604848450439982159111540157939307410328721229713926329334046831720173587249236974523339630036565891415344388403269347755355482010267558029120666672391353119778231074081462998344309980525377722773707707423739, 29748187296456487904874500557836310155656557371050900236754995471110018971948523811144567419032367067617011494296754970385689490917097676543487282945994260980043402596698780863405124066851579921733712136531201772570252486946872946921170277586875344348158652972061260679478405213341513288142643265661196606270240349911882909337806098891659905523142181432866, 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216187142934369712879812661836737467098099295855833823472362250754039338416713963841952440912498396293843687047385015961662264269438769841725327732741251702313454451834272339189021459988815824718842074643223569080407564607018506478745831700208389119136753752881284215900728796337939077136490644937121049797263664432469047670803912345006165148908973895889736], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3\\times C_2^5.C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 8], [6, 8], [8, 2], [12, 12], [24, 2], [48, 2]], 'representations': {'PC': {'code': 7468696736687630335926673635873892123260824075, 'gens': [1, 2, 4, 6], 'pres': [8, -2, -2, -2, -2, -2, 2, -3, -7, 97, 41, 91, 20741, 22669, 141, 46598, 14350, 222, 73735]}, 'Perm': {'d': 22, 'gens': [2439325304712979325, 6307701249, 135854760, 3629520, 167450520, 167450536, 93405312000, 55969571858264064000]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times C_{12}):D_{28}', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
