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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '46080.b', 'ambient_counter': 2, 'ambient_order': 46080, 'ambient_tex': '\\GL(2,5)\\times \\GL(2,\\mathbb{Z}/4)', 'central': False, 'central_factor': False, 'centralizer_order': 384, 'characteristic': True, 'core_order': 240, 'counter': 948, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '46080.b.192.B', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '192.b1.N', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '192.1488', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1488, 'quotient_metabelian': False, 'quotient_nilpotent': False, 'quotient_order': 192, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'C_2\\times \\GL(2,\\mathbb{Z}/4)', 'simple': False, 'solvable': False, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '240.93', 'subgroup_hash': 93, 'subgroup_order': 240, 'subgroup_tex': '\\SL(2,5):C_2', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '46080.b', 'aut_centralizer_order': None, 'aut_label': '192.B', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '120.A', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['64.D', '96.B', '96.E', '96.F', '96.G', '96.H', '96.I'], 'contains': ['384.A'], 'core': '192.B', 'coset_action_label': None, 'count': 1, 'diagramx': [-1, 5396, -1, 5726], 'generators': [104013, 72009, 142649, 8329], 'label': '46080.b.192.B', 'mobius_quo': 0, 'mobius_sub': None, 'normal_closure': '192.B', 'normal_contained_in': ['48.E', '96.B', '96.E'], 'normal_contains': ['384.A', '11520.C'], 'normalizer': '1.a1', 'old_label': '192.b1.N', 'projective_image': '11520.ea', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '192.B', 'subgroup_fusion': None, 'weyl_group': '120.34'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 60, 'aut_gen_orders': [3, 4, 2], 'aut_gens': [[131, 151, 501], [295, 47, 120], [141, 568, 516], [499, 226, 229]], 'aut_group': '240.189', 'aut_hash': 189, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 240, 'aut_permdeg': 7, 'aut_perms': [1110, 360, 415], 'aut_phi_ratio': 3.75, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 30, 1, 1], [3, 20, 1, 1], [4, 1, 2, 1], [4, 30, 1, 1], [5, 12, 2, 1], [6, 20, 1, 1], [10, 12, 2, 1], [12, 20, 2, 1], [20, 12, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_5', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '120.34', 'autcentquo_hash': 34, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 30, 1], [3, 20, 1], [4, 1, 2], [4, 30, 1], [5, 12, 2], [6, 20, 1], [10, 12, 2], [12, 20, 2], [20, 12, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '60.5', 'commutator_count': 1, 'commutator_label': '120.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '60.5'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 93, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 30, 1, 1], [3, 20, 1, 1], [4, 1, 2, 1], [4, 30, 1, 1], [5, 12, 2, 1], [6, 20, 1, 1], [10, 12, 2, 1], [12, 20, 2, 1], [20, 12, 4, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 114, 'exponent': 60, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[2, 0, 4], [4, 0, 2], [6, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '120.35', 'hash': 93, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [5, 5, 2], 'inner_gens': [[131, 295, 511], [107, 151, 576], [146, 226, 501]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 60, 'inner_split': True, 'inner_tex': 'A_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 2], [2, 4], [3, 4], [4, 4], [5, 2], [6, 2]], 'label': '240.93', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SL(2,5):C2', 'ngens': 3, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 11, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 18, 'number_divisions': 11, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 30, 'number_subgroups': 243, 'old_label': None, 'order': 240, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 31], [3, 20], [4, 32], [5, 24], [6, 20], [10, 24], [12, 40], [20, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [126, 126], 'outer_gens': [[464, 444, 30], [131, 151, 129]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [5, 2], [6, 2], [8, 2], [12, 1]], 'representations': {'GLFp': {'d': 2, 'p': 5, 'gens': [131, 151, 501]}, 'Perm': {'d': 24, 'gens': [18038489185576105158077, 493821827582616511657230]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SL(2,5):C_2', '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': '16.10', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [6, 12, 2, 12, 4, 6, 6, 6, 6, 6], 'aut_gens': [[8169, 8301, 8201, 156971, 87829, 141773, 8241, 72367, 24003, 75453], [72001, 50316, 92211, 60883, 148541, 142453, 9601, 140019, 24003, 8081], [105777, 114304, 92211, 158731, 157260, 77601, 108969, 158445, 24003, 43377], [72241, 72109, 8201, 43285, 99747, 13609, 142485, 136059, 24003, 8321], [12809, 154019, 12001, 107377, 5279, 92059, 14401, 78403, 56007, 41937], [75201, 72309, 8201, 108897, 12558, 127255, 9601, 28801, 24003, 142485], [72321, 50316, 92211, 142493, 112585, 73801, 8161, 60241, 56007, 141045], [109137, 134106, 92211, 27367, 1479, 72121, 75453, 45843, 56007, 139365], [8169, 114304, 92211, 126575, 23757, 78601, 142485, 12327, 56007, 8321], [72241, 118104, 92211, 126575, 20477, 9809, 8161, 140091, 56007, 108969], [11209, 134106, 92211, 75441, 4599, 139493, 137925, 8803, 24003, 14401]], 'aut_group': None, 'aut_hash': 3699067435364746117, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 368640, 'aut_permdeg': 108, 'aut_perms': [1221288786334003908171159402030772081619832803704700099358797675562984653211986175320233842730272575177094915212343193175909337265145540507597576767237263052583243144649449959, 97350135673726016670483867823175416809601422410823682438604265180036707794537721531223069968484249488288708018871450896362222726479902984126023110161551572785397200966337955, 833820740053271250035117198473625667064089081484966154321536542662332346213716389802197559887713035079016968875759255068474335771296570253604954317458490250749898526704956372, 114233072719246128462585634043807319123456787864772288443151005108176577884775830537367983967567931757899803406666033062904770596818704470456204604982377432369871826342612689, 540299354806230587788585378184022162827827788141638577725902459561654814059574280517679246078267681816324305980775051925651493571440172677543231723335148071703303201042305722, 620965974405157221587105898540288847922925289692296617442589410719981571224825760463496626522725448033829798183811034447217435394580281489542577766214735041999949400156933450, 160039970614510187511316208810927708006294038797681161763327393676099771994243710503765101194543735416067959996657712300008513491285396059766037580266258081175530565465562558, 522122520021542981600607035484050164860178783216639068372661364903247463440900103867309032398663598019673710589719302664904480949696858199406403326314131512233976106235457701, 468432338969513554670769178666664353878501672867962762089595043880996106371588453143253876899872090438276312454592074626699265087535854165989240805176318367767827373503787126, 1173046592778430719132559748387463613528254336978995103673917452808259284146909621143155930046574514949734185277020353970161666867223310632745243244257106443346721483744729184], 'aut_phi_ratio': 30.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 3, 1, 4], [2, 6, 2, 1], [2, 12, 2, 1], [2, 30, 1, 2], [2, 60, 1, 1], [2, 90, 1, 2], [2, 180, 1, 1], [2, 360, 1, 1], [3, 8, 1, 1], [3, 20, 1, 1], [3, 160, 1, 1], [4, 1, 2, 2], [4, 2, 2, 1], [4, 3, 2, 2], [4, 6, 2, 1], [4, 12, 2, 7], [4, 30, 1, 2], [4, 30, 8, 1], [4, 60, 1, 1], [4, 60, 4, 1], [4, 90, 1, 2], [4, 90, 8, 1], [4, 180, 1, 1], [4, 180, 4, 1], [4, 360, 1, 7], [4, 360, 4, 4], [5, 24, 1, 1], [6, 8, 1, 3], [6, 8, 4, 1], [6, 20, 1, 3], [6, 40, 2, 1], [6, 60, 1, 4], [6, 120, 2, 1], [6, 160, 1, 3], [6, 160, 4, 1], [6, 240, 1, 2], [6, 240, 2, 2], [8, 20, 4, 1], [8, 40, 2, 1], [8, 60, 4, 1], [8, 120, 2, 1], [8, 240, 2, 4], [10, 24, 1, 3], [10, 48, 2, 1], [10, 72, 1, 4], [10, 144, 2, 1], [10, 288, 2, 1], [12, 8, 2, 2], [12, 8, 4, 1], [12, 20, 2, 2], [12, 40, 2, 1], [12, 60, 2, 2], [12, 120, 2, 1], [12, 160, 2, 2], [12, 160, 4, 1], [12, 240, 1, 2], [12, 240, 2, 8], [12, 240, 8, 2], [15, 192, 1, 1], [20, 24, 2, 2], [20, 48, 2, 1], [20, 72, 2, 2], [20, 144, 2, 1], [20, 288, 2, 7], [24, 20, 8, 1], [24, 40, 4, 1], [24, 60, 8, 1], [24, 120, 4, 1], [24, 160, 4, 2], [24, 160, 8, 2], [24, 240, 4, 4], [30, 192, 1, 3], [30, 192, 4, 1], [60, 192, 2, 2], [60, 192, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5\\times S_5\\times C_2^2\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '2880.dv', 'autcentquo_hash': 409072078557532110, 'autcentquo_nilpotent': False, 'autcentquo_order': 2880, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times S_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 3, 4], [2, 6, 2], [2, 12, 2], [2, 30, 2], [2, 60, 1], [2, 90, 2], [2, 180, 1], [2, 360, 1], [3, 8, 1], [3, 20, 1], [3, 160, 1], [4, 1, 4], [4, 2, 2], [4, 3, 4], [4, 6, 2], [4, 12, 14], [4, 30, 10], [4, 60, 5], [4, 90, 10], [4, 180, 5], [4, 360, 23], [5, 24, 1], [6, 8, 7], [6, 20, 3], [6, 40, 2], [6, 60, 4], [6, 120, 2], [6, 160, 7], [6, 240, 6], [8, 20, 4], [8, 40, 2], [8, 60, 4], [8, 120, 2], [8, 240, 8], [10, 24, 3], [10, 48, 2], [10, 72, 4], [10, 144, 2], [10, 288, 2], [12, 8, 8], [12, 20, 4], [12, 40, 2], [12, 60, 4], [12, 120, 2], [12, 160, 8], [12, 240, 34], [15, 192, 1], [20, 24, 4], [20, 48, 2], [20, 72, 4], [20, 144, 2], [20, 288, 14], [24, 20, 8], [24, 40, 4], [24, 60, 8], [24, 120, 4], [24, 160, 24], [24, 240, 16], [30, 192, 7], [60, 192, 8]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '5760.co', 'commutator_count': 1, 'commutator_label': '2880.j', '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', '60.5'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['480.218', 1], ['96.195', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 3, 1, 4], [2, 6, 1, 2], [2, 12, 1, 2], [2, 30, 1, 2], [2, 60, 1, 1], [2, 90, 1, 2], [2, 180, 1, 1], [2, 360, 1, 1], [3, 8, 1, 1], [3, 20, 1, 1], [3, 160, 1, 1], [4, 1, 2, 2], [4, 2, 2, 1], [4, 3, 2, 2], [4, 6, 2, 1], [4, 12, 1, 6], [4, 12, 2, 4], [4, 30, 1, 2], [4, 30, 2, 4], [4, 60, 1, 1], [4, 60, 2, 2], [4, 90, 1, 2], [4, 90, 2, 4], [4, 180, 1, 1], [4, 180, 2, 2], [4, 360, 1, 7], [4, 360, 2, 8], [5, 24, 1, 1], [6, 8, 1, 3], [6, 8, 2, 2], [6, 20, 1, 3], [6, 40, 1, 2], [6, 60, 1, 4], [6, 120, 1, 2], [6, 160, 1, 3], [6, 160, 2, 2], [6, 240, 1, 4], [6, 240, 2, 1], [8, 20, 2, 2], [8, 40, 2, 1], [8, 60, 2, 2], [8, 120, 2, 1], [8, 240, 2, 4], [10, 24, 1, 3], [10, 48, 1, 2], [10, 72, 1, 4], [10, 144, 1, 2], [10, 288, 1, 2], [12, 8, 2, 2], [12, 8, 4, 1], [12, 20, 2, 2], [12, 40, 2, 1], [12, 60, 2, 2], [12, 120, 2, 1], [12, 160, 2, 2], [12, 160, 4, 1], [12, 240, 1, 8], [12, 240, 2, 9], [12, 240, 4, 2], [15, 192, 1, 1], [20, 24, 2, 2], [20, 48, 2, 1], [20, 72, 2, 2], [20, 144, 2, 1], [20, 288, 1, 6], [20, 288, 2, 4], [24, 20, 4, 2], [24, 40, 4, 1], [24, 60, 4, 2], [24, 120, 4, 1], [24, 160, 2, 2], [24, 160, 4, 3], [24, 160, 8, 1], [24, 240, 4, 4], [30, 192, 1, 3], [30, 192, 2, 2], [60, 192, 2, 2], [60, 192, 4, 1]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': None, 'exponent': 120, 'exponents_of_order': [10, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '5760.co', 'hash': 8854831034676951058, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 4, 2, 2, 3, 2, 5, 4, 1, 5], 'inner_gens': [[8169, 8301, 8201, 123295, 20477, 74401, 8161, 72367, 24003, 108969], [8169, 8301, 8201, 156971, 127624, 61963, 8241, 72167, 24003, 75453], [8169, 8301, 8201, 156971, 7639, 141773, 8241, 72367, 24003, 75453], [137773, 8301, 8201, 156971, 87829, 12169, 73693, 41971, 24003, 137925], [73601, 90011, 12001, 156971, 87829, 156131, 11201, 57609, 24003, 45057], [9609, 88311, 8201, 153691, 5279, 141773, 141045, 73807, 24003, 11201], [8089, 8301, 8201, 28887, 151661, 45685, 8241, 72287, 24003, 43377], [8169, 8101, 8201, 30567, 71031, 40965, 8081, 72367, 24003, 41937], [8169, 8301, 8201, 156971, 87829, 141773, 8241, 72367, 24003, 75453], [142653, 8301, 8201, 27527, 148541, 76001, 141045, 46451, 24003, 75453]], 'inner_hash': 6537139331229567448, 'inner_nilpotent': False, 'inner_order': 5760, 'inner_split': None, 'inner_tex': 'C_2\\times S_4\\times S_5', 'inner_used': [1, 2, 4, 5, 8], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 20], [3, 16], [4, 40], [5, 16], [6, 28], [8, 50], [10, 20], [12, 70], [15, 16], [18, 24], [24, 10], [30, 4], [36, 6]], 'label': '46080.b', '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': False, 'name': 'GL(2,5)*GL(2,Z/4)', 'ngens': 10, '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, 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, 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': 152, 'number_characteristic_subgroups': 96, 'number_conjugacy_classes': 336, 'number_divisions': 195, 'number_normal_subgroups': 116, 'number_subgroup_autclasses': 13646, 'number_subgroup_classes': 20933, 'number_subgroups': 1341656, 'old_label': None, 'order': 46080, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 895], [3, 188], [4, 10880], [5, 24], [6, 3236], [8, 2560], [10, 1320], [12, 10144], [15, 192], [20, 4800], [24, 8960], [30, 1344], [60, 1536]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2, 2], 'outer_gen_pows': [53383, 61851, 8001, 75186, 158771, 54301], 'outer_gens': [[72001, 50316, 92211, 60883, 148541, 142453, 9601, 140019, 24003, 8081], [105777, 114304, 92211, 158731, 157260, 77601, 108969, 158445, 24003, 43377], [72241, 72109, 8201, 43285, 99747, 13609, 142485, 136059, 24003, 8321], [12809, 154019, 12001, 107377, 5279, 92059, 14401, 78403, 56007, 41937], [75201, 72309, 8201, 108897, 12558, 127255, 9601, 28801, 24003, 142485], [8169, 114304, 92211, 126575, 23757, 78601, 142485, 12327, 56007, 8321]], 'outer_group': '64.267', 'outer_hash': 267, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 64, 'outer_perms': [1987252695887700205215063048329278735761070605151269409111378011851914488896075475113647, 4279984487754692133726679904469258075958918746186409280759381008378906760689065119180016, 6298565413351235535187384534462565721136803879351282985727179510626611378198031896371200, 8314829610856808383184610145498256263250040713542587318807057103157692609799891633110463, 10330440874710688857744438369768020275239198268616634272172717752048431972891552543657368, 16121957526537530654736414009593691013808258035182399907531691317766273130510103666243970], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6', 'pc_rank': None, 'perfect': False, 'permutation_degree': 32, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [3, 8], [4, 13], [5, 8], [6, 14], [8, 15], [10, 10], [12, 23], [15, 8], [16, 12], [18, 8], [20, 5], [24, 18], [30, 6], [32, 5], [36, 10], [40, 1], [48, 8], [60, 1], [64, 1], [72, 2], [96, 1]], 'representations': {'GLZN': {'d': 2, 'p': 20, 'gens': [8009, 8101, 8201, 25847, 95028, 24927, 8081, 8003, 24003, 9601]}, 'Perm': {'d': 32, 'gens': [26879090983644400076157464728742400, 11662, 5167, 42497847705365890323090029125747200, 60286579449121886431679368817735520, 51742758656347530063266152699326727, 17870848984824952108562641625702400, 8861640035167104415430107167198736, 34601890132546910084515982258724487, 10168084840898584320355983564518400]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 72, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\GL(2,5)\\times \\GL(2,\\mathbb{Z}/4)', 'transitive_degree': None, 'wreath_data': None, 'wreath_product': False}
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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': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 12, 2, 2, 2], 'aut_gens': [[805, 815, 2792, 2821, 2597], [1831, 3865, 2792, 2821, 2597], [2849, 3865, 2792, 2821, 2597], [805, 815, 349, 801, 2597], [3875, 1637, 968, 801, 2821], [805, 2859, 1004, 2821, 2597], [805, 815, 716, 2821, 2597], [805, 2827, 2792, 2821, 2597]], 'aut_group': '768.1088763', 'aut_hash': 1088763, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 768, 'aut_permdeg': 12, 'aut_perms': [167880, 127440, 40723326, 212299932, 16, 7, 90842400], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 6, 2, 1], [2, 12, 2, 1], [3, 8, 1, 1], [4, 12, 2, 3], [6, 8, 1, 1], [6, 8, 2, 1], [6, 8, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6:D_6', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 3, 4], [2, 6, 2], [2, 12, 2], [3, 8, 1], [4, 12, 6], [6, 8, 7]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '48.48', 'commutator_count': 1, 'commutator_label': '24.13', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1488, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['96.195', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 3, 1, 4], [2, 6, 1, 2], [2, 12, 1, 2], [3, 8, 1, 1], [4, 12, 1, 6], [6, 8, 1, 3], [6, 8, 2, 2]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 2520, 'exponent': 12, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '96.226', 'hash': 1488, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 3, 2, 2], 'inner_gens': [[805, 2827, 2792, 2821, 2597], [2849, 815, 349, 801, 2597], [805, 2300, 2792, 2597, 801], [805, 2859, 1004, 2821, 2597], [805, 815, 716, 2821, 2597]], 'inner_hash': 48, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': False, 'inner_tex': 'C_2\\times S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 10], [3, 8], [6, 2]], 'label': '192.1488', 'linC_count': 24, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 8, 'linQ_dim': 5, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2*GL(2,Z/4)', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 16, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 28, 'number_divisions': 26, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 140, 'number_subgroup_classes': 271, 'number_subgroups': 1134, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 55], [3, 8], [4, 72], [6, 56]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [513, 513, 513, 513], 'outer_gens': [[1831, 1853, 2792, 2821, 2597], [805, 815, 968, 2821, 2597], [805, 1853, 349, 801, 2597], [805, 815, 349, 801, 2597]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [288, 7, 127, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [3, 8], [4, 2], [6, 2]], 'representations': {'PC': {'code': 18306840470777316391987088205060, 'gens': [1, 2, 4, 6, 7], 'pres': [7, -2, -2, -2, -2, -3, -2, 2, 85, 36, 570, 80, 571, 3036, 530, 411, 909, 181]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602011816, 330663445672, 706460729520]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [1075659969, 1087456257, 1074808321, 42968769, 1087603969, 1074873345, 1083196673]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [1831, 2853, 1827, 523, 545, 1539, 1628]}, 'Perm': {'d': 10, 'gens': [40446, 289, 55, 288, 45360, 367920, 806400]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times \\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
