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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '384.16398', 'ambient_counter': 16398, 'ambient_order': 384, 'ambient_tex': '(C_6\\times D_8):C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 96, 'counter': 14, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '384.16398.4.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.f1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '96.141', 'subgroup_hash': 141, 'subgroup_order': 96, 'subgroup_tex': 'C_2^3.D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.16398', 'aut_centralizer_order': None, 'aut_label': '4.f1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '48.a1', 'complements': ['96.j1'], 'conjugacy_class_count': 8, 'contained_in': ['2.a1', '2.f1'], 'contains': ['8.c1', '8.d1', '8.e1', '8.o1', '8.o2', '12.i1'], 'core': '4.f1', 'coset_action_label': None, 'count': 8, 'diagramx': None, 'generators': [3, 128, 192, 248, 96, 4], 'label': '384.16398.4.f1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.f1', 'normal_contained_in': ['2.a1', '2.f1'], 'normal_contains': ['8.c1', '8.d1', '8.e1'], 'normalizer': '1.a1', 'old_label': '4.f1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.f1', 'subgroup_fusion': None, 'weyl_group': None}
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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': [2, 6, 4, 2, 2, 2, 2], 'aut_gens': [[1, 2, 8], [49, 2, 40], [1, 70, 56], [25, 2, 8], [5, 6, 12], [5, 2, 8], [49, 50, 8], [5, 6, 8]], 'aut_group': '768.1076197', 'aut_hash': 1076197, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 768, 'aut_permdeg': 15, 'aut_perms': [120, 124106400, 288450408967, 83502720, 88216168320, 16, 7], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 4, 1], [3, 2, 1, 1], [4, 2, 2, 1], [4, 3, 4, 1], [4, 6, 2, 1], [4, 6, 4, 1], [6, 2, 1, 3], [6, 4, 4, 1], [12, 4, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^6:D_6', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [3, 2, 1], [4, 2, 2], [4, 3, 4], [4, 6, 6], [6, 2, 3], [6, 4, 4], [12, 4, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 141, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [3, 2, 1, 1], [4, 2, 1, 2], [4, 3, 2, 2], [4, 6, 2, 3], [6, 2, 1, 3], [6, 4, 1, 4], [12, 4, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 336, 'exponent': 12, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '24.14', 'hash': 141, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 6], 'inner_gens': [[1, 2, 56], [1, 2, 40], [49, 66, 8]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 2]], 'label': '96.141', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 8, 'linQ_dim': 6, 'linQ_dim_count': 44, 'linR_count': 44, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.D6', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 15, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 30, 'number_divisions': 25, 'number_normal_subgroups': 51, 'number_subgroup_autclasses': 52, 'number_subgroup_classes': 94, 'number_subgroups': 178, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 11], [3, 2], [4, 52], [6, 22], [12, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[5, 50, 8], [1, 54, 8], [1, 54, 12], [25, 54, 12]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [16, 23, 120, 11656], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 3]], 'representations': {'PC': {'code': 53630298764499363457, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -2, -2, -2, -3, 31, 1347, 489, 69, 1210, 88, 1163]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101743012388990, 25038659807093174, 58301516036602819]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 27627749610, 74621237504, 75598583751, 102991401878, 47085239376]}, 'Perm': {'d': 11, 'gens': [368767, 1, 16, 11520, 7, 3991680]}}, 'schur_multiplier': [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.D_6', 'transitive_degree': 48, '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': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [6, 8, 8, 4, 12, 8, 4], 'aut_gens': [[1, 2, 8, 16], [5, 70, 300, 112], [197, 167, 57, 276], [197, 359, 56, 272], [5, 3, 109, 305], [5, 226, 249, 112], [5, 99, 57, 20], [197, 259, 296, 368]], 'aut_group': '9216.mt', 'aut_hash': 3235218131844939979, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 98304, 'aut_permdeg': 80, 'aut_perms': [46243369060006925524446340829428325915915910152110009036591481648228696308579366589063017635963177597144191843525114970, 57064224874886747229847700190637197752281265228575954195016679383213177090716189924335273643436962128700250851216595032, 24714695398162198709365857870658985266073088957713743035826708692915235840647209646571487448582095028877179702419084264, 48031298133042776260816508126416639695855661651946929240302013958858713412674843332525025691453754381489942349513150293, 60304193945778028712723644451090976304622558320704466983736114565229793260703681141725195015076299321568566578109035360, 66012231346029570902047057155261135641261093008406328857812022296525130753260475488165571233110328179356432522660645492, 8558276027612217217323902976785849292381369610274973602712266917953001929908792470112834776841435541610141364189841823], 'aut_phi_ratio': 768.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 1, 4, 1], [2, 4, 8, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 6, 8, 1], [4, 12, 8, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [6, 8, 8, 1], [8, 4, 4, 1], [8, 12, 4, 1], [12, 4, 1, 2], [12, 4, 2, 1], [24, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times C_2^6:S_4', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8.C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '48.38', 'autcentquo_hash': 38, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 4, 8], [3, 2, 1], [4, 2, 4], [4, 6, 8], [4, 12, 8], [6, 2, 7], [6, 8, 8], [8, 4, 4], [8, 12, 4], [12, 4, 4], [24, 4, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '48.38', 'commutator_count': 1, 'commutator_label': '12.2', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 16398, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['192.711', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 4, 1, 8], [3, 2, 1, 1], [4, 2, 1, 4], [4, 6, 2, 4], [4, 12, 2, 4], [6, 2, 1, 7], [6, 8, 1, 8], [8, 4, 1, 4], [8, 12, 2, 2], [12, 4, 1, 4], [24, 4, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 65520, 'exponent': 24, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '48.51', 'hash': 16398, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [1, 2, 2, 12], 'inner_gens': [[1, 2, 8, 16], [1, 2, 8, 80], [1, 2, 8, 112], [1, 322, 296, 16]], 'inner_hash': 38, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': False, 'inner_tex': 'S_3\\times D_4', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [4, 16]], 'label': '384.16398', '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': '(C6*D8):C4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 21, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 72, 'number_divisions': 58, 'number_normal_subgroups': 237, 'number_subgroup_autclasses': 154, 'number_subgroup_classes': 608, 'number_subgroups': 1758, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 39], [3, 2], [4, 152], [6, 78], [8, 64], [12, 16], [24, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 4, 4, 2, 4], 'outer_gen_pows': [96, 96, 0, 96, 0], 'outer_gens': [[193, 102, 252, 17], [1, 295, 60, 213], [197, 3, 13, 208], [1, 199, 104, 17], [197, 199, 205, 16]], 'outer_group': '2688.ei', 'outer_hash': 577465834215627728, 'outer_nilpotent': True, 'outer_order': 2048, 'outer_permdeg': 24, 'outer_perms': [109288519158366893358577, 84565253715012735048730, 81137538075865674002980, 109293269718509254859035, 162009384037090840841744], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'F_7\\times C_2^6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 28], [4, 10], [8, 4]], 'representations': {'PC': {'code': 6424936327782374843727278848692220417, 'gens': [1, 2, 4, 5], 'pres': [8, -2, -2, -2, -2, -2, -2, -2, -3, 41, 1612, 588, 116, 3853, 1373, 141, 8974, 166, 8207]}, 'GLZN': {'d': 2, 'p': 40, 'gens': [1344021, 1857019, 193107, 576009, 77793, 64801, 602341, 64401]}, 'Perm': {'d': 17, 'gens': [22505890222081, 4035193031520, 5325060908185, 47102626055544, 69755609548800, 47102626055520, 47102625964800, 3]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_6\\times D_8):C_4', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
