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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1344.8698', 'ambient_counter': 8698, 'ambient_order': 1344, 'ambient_tex': 'C_2\\times C_{12}.D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 2, 'counter': 146, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1344.8698.336.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '336.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': 336, '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': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.8698', 'aut_centralizer_order': None, 'aut_label': '336.f1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '84.d1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['48.f1', '112.g1', '168.f1', '168.h1', '168.j1'], 'contains': ['672.a1'], 'core': '672.a1', 'coset_action_label': None, 'count': 168, 'diagramx': None, 'generators': [15], 'label': '1344.8698.336.f1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.e1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '42.c1', 'old_label': '336.f1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '336.f1', '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': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, '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', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, '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': 3, 'aut_exponent': 168, 'aut_gen_orders': [4, 12, 2, 28, 6, 12, 84], 'aut_gens': [[1, 2, 4, 16], [49, 674, 572, 666], [1067, 682, 678, 280], [105, 2, 4, 218], [201, 2, 350, 16], [873, 2, 686, 410], [483, 10, 228, 408], [721, 2, 460, 698]], 'aut_group': None, 'aut_hash': 7872381701973204557, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2064384, 'aut_permdeg': 160, 'aut_perms': [309160452354559744195578388855877947461223515177783519337112109619495782495972524768017666013071055135532960638192556390407281349554971398553968928334401290360454715565824796393395475109946989472182211065660534788139919911628399370348594321561361535363420452401520924061937765106972328, 288462203500999469806556926460448118525048067510893887044285514986169323408429399113023581505856709599838652924618660655509846534923894202335702873554045726331235567598444711686590170237370552066390375328878706652979163941929362017215064574081685512614551594033899729134014344379446774, 130692968513951563120119219032020082807701344958927259274687080547506846147307482867951624861326671355090484112847353246016175730357259582698104473587469195720885313602037240917702722378320930952319257236512484738523482338201205406783628490107550933536014806606542778935376068688187533, 56354734290960253656513706256062516390017443397008802107757241108931841599435518174839922966218228536648343136446957770298203839652565188579152645358874925372778008230975731014028869446260710538021659902050141286619280544003059371781983760706292622961694692940029793692629304940302319, 14117759917798802951294852804314088780938723442844015991312335578824914647282807647998927795471772414194897913503054569611666105260591753160809999029284595644515653512028921087542909245946710965402748222461675518455078595749968126099891504611902100766463701564737634359896153623092791, 275995592660586714005249505494333860540224761871154491962990961391095266675867072598557312559509923286838930485388321080301203303011890175952685759565815285452509297127734937323716889087583937534691377883085920139875711867500035732388312965107034762511296945720006343636782758726765614, 260244462922001372217221193725706780893193067880938869893956074886387877286562269031320584481976934782362091521138764164993954349071192734578703180614885700288025324378643155139863306178238966332955284393547299646040716813858857969077409501388682856052717844471821044896487251957615907], 'aut_phi_ratio': 5376.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 1, 4, 1], [3, 2, 1, 1], [4, 2, 4, 1], [4, 6, 8, 1], [4, 28, 4, 1], [4, 84, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [7, 2, 3, 1], [12, 4, 4, 1], [12, 28, 8, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 2, 12, 1], [21, 4, 3, 1], [28, 2, 24, 1], [28, 6, 48, 1], [42, 4, 3, 1], [42, 4, 6, 1], [42, 4, 12, 1], [84, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2\\times C_{42}).C_2^6.C_6.C_2^6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 8092891664598236742, 'autcent_nilpotent': True, 'autcent_order': 2048, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8.C_2^3', '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, 7], [3, 2, 1], [4, 2, 4], [4, 6, 8], [4, 28, 4], [4, 84, 4], [6, 2, 7], [7, 2, 3], [12, 4, 4], [12, 28, 8], [14, 2, 21], [21, 4, 3], [28, 2, 24], [28, 6, 48], [42, 4, 21], [84, 4, 24]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '168.50', 'commutator_count': 1, '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': 8698, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['672.533', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [4, 2, 1, 4], [4, 6, 1, 8], [4, 28, 1, 4], [4, 84, 1, 4], [6, 2, 1, 7], [7, 2, 3, 1], [12, 4, 1, 4], [12, 28, 2, 4], [14, 2, 3, 7], [21, 4, 3, 1], [28, 2, 6, 4], [28, 6, 6, 8], [42, 4, 3, 7], [84, 4, 6, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 466830, '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': 8698, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 1, 2, 42], 'inner_gens': [[1, 2, 12, 880], [1, 2, 4, 16], [9, 2, 4, 464], [481, 2, 900, 16]], 'inner_hash': 50, 'inner_nilpotent': False, 'inner_order': 168, 'inner_split': False, 'inner_tex': 'S_3\\times D_{14}', 'inner_used': [1, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 124], [4, 52]], 'label': '1344.8698', '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': 25, 'number_characteristic_subgroups': 26, 'number_conjugacy_classes': 192, 'number_divisions': 76, 'number_normal_subgroups': 224, 'number_subgroup_autclasses': 152, 'number_subgroup_classes': 580, 'number_subgroups': 3700, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 504], [6, 14], [7, 6], [12, 240], [14, 42], [21, 12], [28, 336], [42, 84], [84, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [4, 12, 8, 4, 12], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[347, 10, 686, 18], [673, 682, 14, 498], [1019, 10, 340, 1146], [339, 682, 684, 16], [683, 10, 342, 504]], 'outer_group': None, 'outer_hash': 46930850981611921, 'outer_nilpotent': True, 'outer_order': 12288, 'outer_permdeg': 27, 'outer_perms': [1714377008286120293844001464, 2468877822132100140885628347, 6266640134449978244912162184, 2467559364398385318768971784, 3802844287600260749081925363], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3:(C_2^6.D_4^2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 8], [6, 8], [12, 20], [24, 4]], 'representations': {'PC': {'code': 18069228781767458373841357243406487390655783733225035019, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -7, 5376, 290, 66, 35204, 4660, 116, 19973, 11157, 141, 46598, 7190, 222, 73735]}, 'GLZN': {'d': 2, 'p': 84, 'gens': [593713, 49642529, 7705165, 42082055, 1042546, 17588257, 24300905, 42577765]}, 'Perm': {'d': 24, 'gens': [1126440057374177817727, 47338370961853, 120, 69319678176000, 91811278137623, 91811278137600, 45360, 28104890400830251008000]}}, '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': 1344, 'wreath_data': None, 'wreath_product': False}
