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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '768.87077', 'ambient_counter': 87077, 'ambient_order': 768, 'ambient_tex': '(C_2^3\\times C_{12}).D_4', 'central': True, 'central_factor': False, 'centralizer_order': 768, 'characteristic': True, 'core_order': 1, 'counter': 1118, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '768.87077.768.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '768.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '768.87077', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 87077, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 768, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': '(C_2^3\\times C_{12}).D_4', 'simple': False, 'solvable': True, 'special_labels': ['PC', 'U4', 'D2', 'C9'], 'split': True, 'standard_generators': False, 'stem': True, 'subgroup': '1.1', 'subgroup_hash': 1, 'subgroup_order': 1, 'subgroup_tex': 'C_1', 'supersolvable': True, 'sylow': 1}
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gps_subgroup_data • Show schema
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{'ambient': '768.87077', 'aut_centralizer_order': None, 'aut_label': '768.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1.a1', 'complements': ['1.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['256.a1.a1', '384.a1.a1', '384.b1.a1', '384.c1.a1', '384.d1.a1', '384.d1.b1', '384.e1.a1', '384.f1.a1', '384.g1.a1', '384.g1.b1', '384.h1.a1', '384.i1.a1', '384.j1.a1', '384.j1.b1', '384.k1.a1', '384.l1.a1', '384.m1.a1', '384.m1.b1'], 'contains': [], 'core': '768.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [], 'label': '768.87077.768.a1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '768.a1.a1', 'normal_contained_in': ['256.a1.a1', '384.c1.a1', '384.b1.a1', '384.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '768.a1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '768.a1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1.1', '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': [], '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]], '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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 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': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 2, 2, 12, 2, 2, 4, 12, 4], 'aut_gens': [[1, 4, 8, 32, 128], [407, 20, 392, 500, 144], [143, 84, 408, 240, 640], [83, 388, 88, 416, 640], [205, 68, 392, 564, 144], [209, 20, 88, 176, 640], [81, 20, 72, 496, 128], [85, 388, 472, 52, 656], [533, 388, 408, 564, 144], [459, 452, 88, 36, 656]], 'aut_group': None, 'aut_hash': 2194389629022892761, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24576, 'aut_permdeg': 112, 'aut_perms': [6297349625138305749064321763609333047936063757586156202324073079326167305233514088528470783436557414248340478668818011043524625066349005887056785565216770213632849606476378282111617, 15477103798614171299642284623931294127149881063789807030179434445705061319728199007537437063712082460094210044923645610475250185772312943735907478910776381833559926797831933547020372, 138929584290816069610901339307551556302042000593032327129135187067267480219457202792813970854217804339825166158647539465019405931480219604018774088665204060499130454039536476487839814, 170840274763998461595430561049655134740770851563699572426415747871008809071913161939382795774191314279940664550916172890870519374940529974418740489901032724158245124139555126613096937, 87947316178431297086794998710195912457765361176781011653444506335240132704932545789284556224556933733686291986868041272459413506253970225267943402191258919526604854845682381209773175, 90399424799967029650800697871341592510572888334022590072788977129462826027910661240958012433599994037194016151788657334534870486187261607260728581525721476604670994333142797026463154, 177295195840445626581622776894764975325700940289518188724142975062842780325821191034103993795422365832233101009615602407197269600781176238462758794081974972861240238646242469247160894, 18883615984931150782336412974587088129024349261970549472373016104843352104862364467784875147206142049282814928867624927982950653206552806150396355275993829237672630466226318671568439, 134296079448967589780408932172365805950429517004388973506555720528787467052375505288289114077796007896627961092298728937266285237598312131270228653163441928688897622153911547965416647], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 2, 2, 1], [2, 4, 1, 3], [2, 4, 2, 2], [2, 8, 1, 1], [2, 24, 2, 1], [3, 2, 1, 1], [4, 4, 2, 1], [4, 8, 1, 1], [4, 8, 8, 1], [4, 24, 2, 1], [4, 48, 1, 2], [6, 2, 1, 3], [6, 2, 4, 1], [6, 4, 1, 2], [6, 4, 2, 3], [6, 4, 4, 1], [6, 8, 2, 2], [8, 48, 4, 1], [12, 8, 2, 2], [12, 8, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^8.C_2^5)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '384.18491', 'autcentquo_hash': 18491, 'autcentquo_nilpotent': False, 'autcentquo_order': 384, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^5:D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [2, 4, 7], [2, 8, 1], [2, 24, 2], [3, 2, 1], [4, 4, 2], [4, 8, 9], [4, 24, 2], [4, 48, 2], [6, 2, 7], [6, 4, 12], [6, 8, 4], [8, 48, 4], [12, 8, 20]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '192.513', 'commutator_count': 1, 'commutator_label': '48.52', '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': 87077, '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, 4], [2, 4, 1, 7], [2, 8, 1, 1], [2, 24, 1, 2], [3, 2, 1, 1], [4, 4, 1, 2], [4, 8, 1, 1], [4, 8, 2, 4], [4, 24, 1, 2], [4, 48, 1, 2], [6, 2, 1, 3], [6, 2, 2, 2], [6, 4, 1, 2], [6, 4, 2, 5], [6, 8, 1, 2], [6, 8, 2, 1], [8, 48, 2, 2], [12, 8, 1, 2], [12, 8, 2, 1], [12, 8, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5376, 'exponent': 24, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '32.46', 'frattini_quotient': '24.14', 'hash': 87077, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 2, 2, 2, 6], 'inner_gens': [[1, 404, 72, 484, 656], [401, 4, 24, 32, 128], [65, 20, 8, 480, 128], [453, 4, 456, 32, 640], [273, 4, 8, 288, 128]], 'inner_hash': 513, 'inner_nilpotent': False, 'inner_order': 192, 'inner_split': None, 'inner_tex': 'C_2^3.D_{12}', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 44], [4, 16], [8, 5]], 'label': '768.87077', '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': '(C2^3*C12).D4', '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': 37, 'number_characteristic_subgroups': 69, 'number_conjugacy_classes': 81, 'number_divisions': 54, 'number_normal_subgroups': 111, 'number_subgroup_autclasses': 658, 'number_subgroup_classes': 1118, 'number_subgroups': 6180, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 95], [3, 2], [4, 224], [6, 94], [8, 192], [12, 160]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [70, 384, 452, 2, 0, 256], 'outer_gens': [[5, 404, 408, 500, 656], [467, 404, 72, 112, 640], [599, 20, 456, 180, 656], [81, 404, 24, 420, 144], [83, 388, 88, 416, 640], [157, 84, 456, 244, 144]], 'outer_group': '128.2320', 'outer_hash': 2320, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 64, 'outer_perms': [52702425846698886940321581434951218761010675276314658729011082179790112848822140946135200, 12721693891497978669412471695142931445773478890355664669164233140423870393534686800521128, 102483329167724258507634763783200364280051197544841975352021509816271427642167387909919486, 102388827578402358087753325881466087759411334091780430445327486080980710651575044237355478, 4185591064254038755246484871295595877966671539262515487948425357000534363013673798559560, 50687755791566670366009375847040936752325876924377872213215278879755717689582439761493643], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 20], [8, 9], [16, 1]], 'representations': {'PC': {'code': '172244333227204370699791055482945763366810119378338522682881', 'gens': [1, 3, 4, 6, 8], 'pres': [9, 2, 2, 2, 2, 2, 2, 2, 2, 3, 18, 154, 10910, 281, 2595, 237, 102, 26141, 11246, 3272, 158, 47239, 1492, 214, 41480, 1349]}, 'Perm': {'d': 15, 'gens': [106948356505, 175964181864, 1680, 295076597760, 744, 388481909760, 442680940800, 448908001920, 3]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2^3\\times C_{12}).D_4', 'transitive_degree': 96, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 2, 2, 12, 2, 2, 4, 12, 4], 'aut_gens': [[1, 4, 8, 32, 128], [407, 20, 392, 500, 144], [143, 84, 408, 240, 640], [83, 388, 88, 416, 640], [205, 68, 392, 564, 144], [209, 20, 88, 176, 640], [81, 20, 72, 496, 128], [85, 388, 472, 52, 656], [533, 388, 408, 564, 144], [459, 452, 88, 36, 656]], 'aut_group': None, 'aut_hash': 2194389629022892761, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24576, 'aut_permdeg': 112, 'aut_perms': [6297349625138305749064321763609333047936063757586156202324073079326167305233514088528470783436557414248340478668818011043524625066349005887056785565216770213632849606476378282111617, 15477103798614171299642284623931294127149881063789807030179434445705061319728199007537437063712082460094210044923645610475250185772312943735907478910776381833559926797831933547020372, 138929584290816069610901339307551556302042000593032327129135187067267480219457202792813970854217804339825166158647539465019405931480219604018774088665204060499130454039536476487839814, 170840274763998461595430561049655134740770851563699572426415747871008809071913161939382795774191314279940664550916172890870519374940529974418740489901032724158245124139555126613096937, 87947316178431297086794998710195912457765361176781011653444506335240132704932545789284556224556933733686291986868041272459413506253970225267943402191258919526604854845682381209773175, 90399424799967029650800697871341592510572888334022590072788977129462826027910661240958012433599994037194016151788657334534870486187261607260728581525721476604670994333142797026463154, 177295195840445626581622776894764975325700940289518188724142975062842780325821191034103993795422365832233101009615602407197269600781176238462758794081974972861240238646242469247160894, 18883615984931150782336412974587088129024349261970549472373016104843352104862364467784875147206142049282814928867624927982950653206552806150396355275993829237672630466226318671568439, 134296079448967589780408932172365805950429517004388973506555720528787467052375505288289114077796007896627961092298728937266285237598312131270228653163441928688897622153911547965416647], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 2, 2, 1], [2, 4, 1, 3], [2, 4, 2, 2], [2, 8, 1, 1], [2, 24, 2, 1], [3, 2, 1, 1], [4, 4, 2, 1], [4, 8, 1, 1], [4, 8, 8, 1], [4, 24, 2, 1], [4, 48, 1, 2], [6, 2, 1, 3], [6, 2, 4, 1], [6, 4, 1, 2], [6, 4, 2, 3], [6, 4, 4, 1], [6, 8, 2, 2], [8, 48, 4, 1], [12, 8, 2, 2], [12, 8, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^8.C_2^5)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '384.18491', 'autcentquo_hash': 18491, 'autcentquo_nilpotent': False, 'autcentquo_order': 384, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^5:D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [2, 4, 7], [2, 8, 1], [2, 24, 2], [3, 2, 1], [4, 4, 2], [4, 8, 9], [4, 24, 2], [4, 48, 2], [6, 2, 7], [6, 4, 12], [6, 8, 4], [8, 48, 4], [12, 8, 20]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '192.513', 'commutator_count': 1, 'commutator_label': '48.52', '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': 87077, '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, 4], [2, 4, 1, 7], [2, 8, 1, 1], [2, 24, 1, 2], [3, 2, 1, 1], [4, 4, 1, 2], [4, 8, 1, 1], [4, 8, 2, 4], [4, 24, 1, 2], [4, 48, 1, 2], [6, 2, 1, 3], [6, 2, 2, 2], [6, 4, 1, 2], [6, 4, 2, 5], [6, 8, 1, 2], [6, 8, 2, 1], [8, 48, 2, 2], [12, 8, 1, 2], [12, 8, 2, 1], [12, 8, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5376, 'exponent': 24, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '32.46', 'frattini_quotient': '24.14', 'hash': 87077, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 2, 2, 2, 6], 'inner_gens': [[1, 404, 72, 484, 656], [401, 4, 24, 32, 128], [65, 20, 8, 480, 128], [453, 4, 456, 32, 640], [273, 4, 8, 288, 128]], 'inner_hash': 513, 'inner_nilpotent': False, 'inner_order': 192, 'inner_split': None, 'inner_tex': 'C_2^3.D_{12}', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 44], [4, 16], [8, 5]], 'label': '768.87077', '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': '(C2^3*C12).D4', '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': 37, 'number_characteristic_subgroups': 69, 'number_conjugacy_classes': 81, 'number_divisions': 54, 'number_normal_subgroups': 111, 'number_subgroup_autclasses': 658, 'number_subgroup_classes': 1118, 'number_subgroups': 6180, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 95], [3, 2], [4, 224], [6, 94], [8, 192], [12, 160]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [70, 384, 452, 2, 0, 256], 'outer_gens': [[5, 404, 408, 500, 656], [467, 404, 72, 112, 640], [599, 20, 456, 180, 656], [81, 404, 24, 420, 144], [83, 388, 88, 416, 640], [157, 84, 456, 244, 144]], 'outer_group': '128.2320', 'outer_hash': 2320, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 64, 'outer_perms': [52702425846698886940321581434951218761010675276314658729011082179790112848822140946135200, 12721693891497978669412471695142931445773478890355664669164233140423870393534686800521128, 102483329167724258507634763783200364280051197544841975352021509816271427642167387909919486, 102388827578402358087753325881466087759411334091780430445327486080980710651575044237355478, 4185591064254038755246484871295595877966671539262515487948425357000534363013673798559560, 50687755791566670366009375847040936752325876924377872213215278879755717689582439761493643], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 20], [8, 9], [16, 1]], 'representations': {'PC': {'code': '172244333227204370699791055482945763366810119378338522682881', 'gens': [1, 3, 4, 6, 8], 'pres': [9, 2, 2, 2, 2, 2, 2, 2, 2, 3, 18, 154, 10910, 281, 2595, 237, 102, 26141, 11246, 3272, 158, 47239, 1492, 214, 41480, 1349]}, 'Perm': {'d': 15, 'gens': [106948356505, 175964181864, 1680, 295076597760, 744, 388481909760, 442680940800, 448908001920, 3]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2^3\\times C_{12}).D_4', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
