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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '256.5425', 'ambient_counter': 5425, 'ambient_order': 256, 'ambient_tex': 'C_{16}.D_8', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 16, 'counter': 40, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '256.5425.8.m1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.m1.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': 8, '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': '32.16', 'subgroup_hash': 16, 'subgroup_order': 32, 'subgroup_tex': 'C_2\\times C_{16}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.5425', 'aut_centralizer_order': 64, 'aut_label': '8.m1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '16.10', 'aut_weyl_index': 256, 'centralizer': '8.m1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.f1.a1'], 'contains': ['16.f1.a1', '16.l1.a1', '16.t1.a1'], 'core': '16.f1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [5538, -1, 3553, -1, 5538, -1, 3553, -1], 'generators': [1, 16], 'label': '256.5425.8.m1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.f1.a1', 'old_label': '8.m1.a1', 'projective_image': '128.399', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.m1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '32.16', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 2, 2, 4, 2], 'aut_gens': [[1, 2], [17, 15], [1, 3], [1, 30], [1, 10], [1, 14]], 'aut_group': '32.48', 'aut_hash': 48, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 10, 'aut_perms': [455286, 374406, 1270441, 859248, 1], '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, 4, 2], [16, 1, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4:C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.48', 'autcent_hash': 48, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4:C_2^2', '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, 3], [4, 1, 4], [8, 1, 8], [16, 1, 16]], 'center_label': '32.16', 'center_order': 32, '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', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 16, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['16.1', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [8, 1, 4, 2], [16, 1, 8, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 16, 'exponents_of_order': [5], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '4.2', 'hash': 16, '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, 32]], 'label': '32.16', 'linC_count': 192, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 4, 'linQ_dim': 9, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C16', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 32, 'number_divisions': 10, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 14, 'number_subgroups': 14, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4], [8, 8], [16, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4, 2, 2, 4, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[17, 15], [1, 3], [1, 30], [1, 10], [1, 14]], 'outer_group': '32.48', 'outer_hash': 48, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [455286, 374406, 1270441, 859248, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4:C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 18, 'pgroup': 2, 'primary_abelian_invariants': [2, 16], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 2], [8, 2]], 'representations': {'PC': {'code': 17891342, 'gens': [1, 2], 'pres': [5, -2, 2, -2, -2, -2, 26, 42, 58]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [19665, 14742]}, 'Perm': {'d': 18, 'gens': [20916435456000, 355687428096000, 9703614452976, 4097506710982, 1313941673647]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 16], '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_{16}', 'transitive_degree': 32, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 16, 'aut_gen_orders': [8, 8, 8, 8, 16, 4, 8], 'aut_gens': [[1, 2, 16], [5, 42, 144], [141, 162, 240], [133, 114, 240], [133, 10, 176], [5, 254, 16], [137, 70, 176], [129, 122, 48]], 'aut_group': None, 'aut_hash': 593374907856038680, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 72, 'aut_perms': [57928636994571521390973387799190076155968798548815301727971947107484578613389877585237903938958936697911, 41565968657929463655986870976336697777171849993075727360710377010225163304534702052751454384908618753677, 38980087093771576531551636782238692310143685396098822766726995649533230668394189212982337776103902677957, 53251987600557728535913730770371404344698787061657782796975199036538114664865453826261305883502247673039, 10190515334466711180818049838637739344314102331070185268445186284447004143516330839487979663788357420946, 6280456480583323210825597938322557465060899460581566242918850099392147454349172323049578383881855499999, 32924853948485143239011975444425746518308227167587955364971548786179355314111635468992720832979994574532], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 8, 1, 1], [4, 2, 1, 2], [4, 4, 2, 1], [4, 8, 1, 1], [4, 32, 2, 1], [8, 2, 2, 2], [8, 4, 1, 2], [8, 8, 2, 1], [16, 2, 4, 1], [16, 4, 2, 1], [16, 4, 4, 1], [16, 8, 4, 1], [16, 16, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2.C_4^3.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': '512.10483248', 'autcentquo_hash': 5580842273875875444, 'autcentquo_nilpotent': True, 'autcentquo_order': 512, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4^2:C_2^3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 8, 1], [4, 2, 2], [4, 4, 2], [4, 8, 1], [4, 32, 2], [8, 2, 4], [8, 4, 2], [8, 8, 2], [16, 2, 4], [16, 4, 6], [16, 8, 4], [16, 16, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '128.399', 'commutator_count': 1, 'commutator_label': '32.3', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 5425, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 8, 1, 1], [4, 2, 1, 2], [4, 4, 1, 2], [4, 8, 1, 1], [4, 32, 1, 2], [8, 2, 2, 2], [8, 4, 1, 2], [8, 8, 2, 1], [16, 2, 4, 1], [16, 4, 2, 1], [16, 4, 4, 1], [16, 8, 4, 1], [16, 16, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1344, 'exponent': 16, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, -1, 8]], 'familial': False, 'frattini_label': '32.3', 'frattini_quotient': '8.5', 'hash': 5425, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 8, 8], 'inner_gens': [[1, 14, 16], [133, 2, 240], [1, 34, 16]], 'inner_hash': 399, 'inner_nilpotent': True, 'inner_order': 128, 'inner_split': False, 'inner_tex': 'C_8:D_8', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 32, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 18], [4, 11]], 'label': '256.5425', 'linC_count': 8, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 2, 'linQ_dim': 32, 'linQ_dim_count': 2, 'linR_count': 8, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C16.D8', 'ngens': 3, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 19, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 37, 'number_divisions': 22, 'number_normal_subgroups': 37, 'number_subgroup_autclasses': 82, 'number_subgroup_classes': 107, 'number_subgroups': 319, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 11], [4, 84], [8, 32], [16, 128]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [16, 0, 64, 0], 'outer_gens': [[1, 18, 16], [129, 2, 240], [1, 10, 16], [1, 2, 80]], 'outer_group': '32.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 5040, 120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 64, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 6], [8, 2], [16, 2]], 'representations': {'PC': {'code': 10759575582201494380796581920767411198, 'gens': [1, 2, 5], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 225, 41, 3362, 66, 4355, 539, 4812, 116, 5389, 141, 5390, 166]}, 'Perm': {'d': 64, 'gens': [66450134624445933678799560050331607746391532584685435254529204790382042061733888000000000, 44320550212058861796589790906002973487639595198274528990377915885996743223267447294622038, 16116754647525886103169130832883889877597823742414793216380478825221577993660115377372856, 8058377180213801897373328225453205837655325156463172012191076551884877043098396222780856, 12087040917006856825820464386751442449456779291427783372556125414865848445276683553449656, 4028664439651445760304855308136396429713341927997617409871981318782482722497452481942982, 6010765313484600225807159681127960740186976874672868603891438014304108462838676947041862, 1983116034341975454789449189498620955194254460449807567074683265838922666583379392681647]}}, 'schur_multiplier': [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_{16}.D_8', 'transitive_degree': 64, 'wreath_data': None, 'wreath_product': False}
