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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '12352.1674', 'ambient_counter': 1674, 'ambient_order': 12352, 'ambient_tex': 'C_{772}:C_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 64, 'characteristic': False, 'core_order': 2, 'counter': 36, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '12352.1674.772.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '772.c1.a1', 'outer_equivalence': False, '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': 772, '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': '16.5', 'subgroup_hash': 5, 'subgroup_order': 16, 'subgroup_tex': 'C_2\\times C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '12352.1674', 'aut_centralizer_order': None, 'aut_label': '772.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '193.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.c1.a1', '386.a1.a1', '386.b1.a1', '386.b1.b1'], 'contains': ['1544.b1.a1', '1544.e1.a1', '1544.e1.b1'], 'core': '6176.a1.a1', 'coset_action_label': None, 'count': 193, 'diagramx': [8564, -1, 6304, -1, 2548, -1, 6654, -1], 'generators': [2, 6176], 'label': '12352.1674.772.c1.a1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '193.a1.a1', 'old_label': '772.c1.a1', 'projective_image': '6176.246', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '772.c1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.5', '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], 'aut_gens': [[1, 2], [1, 3], [9, 14], [1, 6], [1, 10]], 'aut_group': '16.11', 'aut_hash': 11, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 6, 'aut_perms': [126, 55, 289, 288], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 2, 2], [8, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.11', 'autcent_hash': 11, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times D_4', '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, 3], [4, 1, 4], [8, 1, 8]], 'center_label': '16.5', 'center_order': 16, 'central_product': True, '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', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [8, 1, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 8, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '4.2', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 16]], 'label': '16.5', 'linC_count': 48, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 4, 'linQ_dim': 5, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C8', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 16, 'number_divisions': 8, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 11, 'number_subgroups': 11, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4], [8, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 3], [9, 14], [1, 6], [1, 10]], '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': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 2]], 'representations': {'PC': {'code': 9222, 'gens': [1, 2], 'pres': [4, -2, 2, -2, -2, 21, 34]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706304316034]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [19665, 44226]}, 'Perm': {'d': 10, 'gens': [40176, 362880, 16582, 5167]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_8', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '64.26', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 37056, 'aut_gen_orders': [48, 24, 64, 48], 'aut_gens': [[1, 16], [4273, 3856], [5809, 9008], [3601, 464], [529, 8816]], 'aut_group': None, 'aut_hash': 2013647253270866901, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 296448, 'aut_permdeg': 772, 'aut_perms': 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'aut_phi_ratio': 48.25, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 193, 1, 2], [4, 1, 2, 1], [4, 193, 1, 4], [4, 193, 2, 3], [8, 193, 2, 8], [16, 193, 4, 8], [193, 16, 12, 1], [386, 16, 12, 1], [772, 16, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{386}.C_{96}.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 37056, 'autcentquo_group': '37056.a', 'autcentquo_hash': 1432757702296672986, 'autcentquo_nilpotent': False, 'autcentquo_order': 37056, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{193}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 193, 2], [4, 1, 2], [4, 193, 10], [8, 193, 16], [16, 193, 32], [193, 16, 12], [386, 16, 12], [772, 16, 24]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '3088.54', 'commutator_count': 1, 'commutator_label': '193.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '193.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1674, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3088.54', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 193, 1, 2], [4, 1, 2, 1], [4, 193, 2, 5], [8, 193, 4, 4], [16, 193, 8, 4], [193, 16, 12, 1], [386, 16, 12, 1], [772, 16, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 192, 'exponent': 3088, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 193], 'factors_of_order': [2, 193], 'faithful_reps': [[16, 0, 24]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '6176.246', 'hash': 1674, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 3088, 'inner_gen_orders': [16, 193], 'inner_gens': [[1, 12304], [65, 16]], 'inner_hash': 54, 'inner_nilpotent': False, 'inner_order': 3088, 'inner_split': False, 'inner_tex': 'C_{193}:C_{16}', 'inner_used': [1, 2], 'irrC_degree': 16, 'irrQ_degree': 384, 'irrQ_dim': 384, 'irrR_degree': 32, 'irrep_stats': [[1, 64], [16, 48]], 'label': '12352.1674', 'linC_count': 24, 'linC_degree': 16, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 194, 'linQ_degree_count': 8, 'linQ_dim': 194, 'linQ_dim_count': 8, 'linR_count': 384, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C772:C16', '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': 31, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 112, 'number_divisions': 21, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 44, 'number_subgroup_classes': 58, 'number_subgroups': 5050, 'old_label': None, 'order': 12352, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 387], [4, 1932], [8, 3088], [16, 6176], [193, 192], [386, 192], [772, 384]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 4, 12], 'outer_gen_pows': [0, 5, 12], 'outer_gens': [[1, 12336], [1, 7312], [3089, 11408]], 'outer_group': '96.165', 'outer_hash': 165, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [767760, 2160, 4032004], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_{12}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 197, 'pgroup': 0, 'primary_abelian_invariants': [4, 16], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 4], [8, 4], [192, 2], [384, 1]], 'representations': {'PC': {'code': '72407528335999452377277843788303020815094084830022278204159326426892925831612632776520499391', 'gens': [1, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -193, 14, 36, 58, 430644, 2531, 11358, 26975, 102, 514757, 6060, 27235, 64706, 124, 595846, 14125, 63524, 75291]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [7189251, 1365920873, 582313729]}, 'Perm': {'d': 197, 'gens': [5290273345984864897683944123082956631220066364900253396726259059766269559200235413715177147698004284025594961028822560923834716647335335646782683178127927701850149621311947644736352530825831559089639200063438532790427221928341897546834098797438127745470625459344068218749828555758253110619759250004403483806498954818604823553161926087080150002071074875160801725560, 9, 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2592381132973149765779289246531185296419382784883920013901222755849351293628592008151691890954402788253039663753020082609107439334904412326282239603261948437302473854804655963712088645187779112731793528834357671948713737992400807244149705391268119111136677134958116230430763311877262318184849472983323284357135732667011753861924031209391791363940984026715737429240, 520972671180291378873378228799679326959717638111606330328292901165986473129410184711055071299345784828690369857414184422249346580994648235923978424589548271404017967409610673350600057073095348109955036142474517622285973437735974016368518172307132793835132086153054465008777445548393143686583542501166362214033044187300451811977217876794123160958270890813465818123440]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 16], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{772}:C_{16}', 'transitive_degree': 772, 'wreath_data': None, 'wreath_product': False}
