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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1856.389', 'ambient_counter': 389, 'ambient_order': 1856, 'ambient_tex': 'C_{29}:Q_{64}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 32, 'counter': 14, 'cyclic': False, 'direct': None, 'hall': 2, 'label': '1856.389.29.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '29.a1.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': 29, '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': '64.54', 'subgroup_hash': 54, 'subgroup_order': 64, 'subgroup_tex': 'Q_{64}', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '1856.389', 'aut_centralizer_order': 28, 'aut_label': '29.a1', 'aut_quo_index': None, 'aut_stab_index': 29, 'aut_weyl_group': '256.5055', 'aut_weyl_index': 812, 'centralizer': '928.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['58.a1.a1', '58.b1.a1', '58.c1.a1'], 'core': '58.a1.a1', 'coset_action_label': None, 'count': 29, 'diagramx': [8651, -1, 761, -1, 6574, -1, 3426, -1], 'generators': [1, 2], 'label': '1856.389.29.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '29.a1.a1', 'old_label': '29.a1.a1', 'projective_image': '928.33', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '29.a1.a1', 'subgroup_fusion': None, 'weyl_group': '32.18'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 32, 'aut_gen_orders': [2, 8, 32], 'aut_gens': [[1, 2], [31, 62], [21, 6], [51, 34]], 'aut_group': '512.58302', 'aut_hash': 7772715459544409331, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 32, 'aut_perms': [209110625063666949217196394943844509, 166831302240608821283181734550591682, 192584446486825221723391051205525821], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 16, 2, 1], [8, 2, 2, 1], [16, 2, 4, 1], [32, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{32}:C_8', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 16, 'autcentquo_group': '128.913', 'autcentquo_hash': 913, 'autcentquo_nilpotent': True, 'autcentquo_order': 128, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{16}:C_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 16, 2], [8, 2, 2], [16, 2, 4], [32, 2, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '32.18', 'commutator_count': 1, 'commutator_label': '16.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 54, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 16, 1, 2], [8, 2, 2, 1], [16, 2, 4, 1], [32, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 32, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, -1, 8]], 'familial': True, 'frattini_label': '16.1', 'frattini_quotient': '4.2', 'hash': 54, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 16, 'inner_gen_orders': [2, 16], 'inner_gens': [[1, 62], [5, 2]], 'inner_hash': 18, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'D_{16}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 32, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 15]], 'label': '64.54', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 32, 'linQ_dim_count': 1, 'linR_count': 8, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'Q64', 'ngens': 2, 'nilpotency_class': 5, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 19, 'number_divisions': 8, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 11, 'number_subgroup_classes': 15, 'number_subgroups': 37, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 34], [8, 4], [16, 8], [32, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 8, 'outer_gen_orders': [2, 8], 'outer_gen_pows': [2, 0], 'outer_gens': [[3, 2], [1, 10]], 'outer_group': '16.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 10, 'outer_perms': [362880, 6057], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_8', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 64, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1], [8, 1], [16, 1]], 'representations': {'PC': {'code': 309504273051162771967, 'gens': [1, 2], 'pres': [6, -2, 2, -2, -2, -2, -2, 192, 745, 31, 1082, 50, 1347, 69, 1444, 88]}, 'Lie': [{'d': 2, 'q': 31, 'gens': [896293, 653851, 893760, 405374], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 31, 'gens': [893666, 687030]}, 'Perm': {'d': 64, 'gens': [66450134624445933678799560050331607746391532584685435254529204790382042061733888000000000, 34216625329393954566794504615428400130324458319121250657957939996263067413000027270329600, 24175131827739636274127375045840733860693888662420563914564207161518320707982436785337600, 12087040917006882684584735480871444149038188995887599735100184001816510068360415118652856, 6010765313484612917937180565556780809330360821355660294198753718800751569568892186006982, 1983116034341975454789449189498620955194254460449807567074683265838922666583379392681647]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_{64}', 'transitive_degree': 64, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 3248, 'aut_gen_orders': [232, 56, 56, 16, 28], 'aut_gens': [[1, 2, 64], [1773, 1750, 64], [1497, 1510, 640], [1461, 1414, 1472], [709, 706, 1792], [593, 622, 320]], 'aut_group': None, 'aut_hash': 7647928247098134465, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 207872, 'aut_permdeg': 480, 'aut_perms': 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'aut_phi_ratio': 232.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 16, 1, 1], [4, 464, 1, 1], [8, 2, 2, 1], [16, 2, 4, 1], [29, 2, 14, 1], [32, 58, 8, 1], [58, 2, 14, 1], [116, 4, 14, 1], [116, 16, 28, 1], [232, 4, 28, 1], [464, 4, 56, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{232}.C_{14}.C_2^3.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1624, 'autcentquo_group': None, 'autcentquo_hash': 3014453250529510555, 'autcentquo_nilpotent': False, 'autcentquo_order': 51968, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{116}.C_{14}.C_2^4.C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 16, 1], [4, 464, 1], [8, 2, 2], [16, 2, 4], [29, 2, 14], [32, 58, 8], [58, 2, 14], [116, 4, 14], [116, 16, 28], [232, 4, 28], [464, 4, 56]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '928.33', 'commutator_count': 1, 'commutator_label': '464.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '29.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 389, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 16, 1, 1], [4, 464, 1, 1], [8, 2, 2, 1], [16, 2, 4, 1], [29, 2, 14, 1], [32, 58, 8, 1], [58, 2, 14, 1], [116, 4, 14, 1], [116, 16, 28, 1], [232, 4, 28, 1], [464, 4, 56, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 928, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 7, 29], 'factors_of_order': [2, 29], 'faithful_reps': [[4, -1, 56]], 'familial': False, 'frattini_label': '16.1', 'frattini_quotient': '116.4', 'hash': 389, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 464, 'inner_gen_orders': [2, 16, 29], 'inner_gens': [[1, 62, 1792], [5, 2, 1792], [129, 130, 64]], 'inner_hash': 33, 'inner_nilpotent': False, 'inner_order': 928, 'inner_split': True, 'inner_tex': 'C_{29}:D_{16}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 224, 'irrQ_dim': 448, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 71], [4, 98]], 'label': '1856.389', 'linC_count': 504, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 44, 'linQ_degree_count': 2, 'linQ_dim': 60, 'linQ_dim_count': 2, 'linR_count': 224, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C29:Q64', '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': 14, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 173, 'number_divisions': 14, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 30, 'number_subgroups': 550, 'old_label': None, 'order': 1856, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 1], [4, 482], [8, 4], [16, 8], [29, 28], [32, 464], [58, 28], [116, 504], [232, 112], [464, 224]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 56, 'outer_gen_orders': [4, 56], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 18, 1088], [33, 42, 640]], 'outer_group': '224.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 224, 'outer_permdeg': 19, 'outer_perms': [13537045057536000, 562834138473], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times C_{56}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 93, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1], [8, 1], [16, 1], [28, 2], [56, 2], [112, 1], [224, 1]], 'representations': {'PC': {'code': 23729776768461155768980057858218008019201329493327899, 'gens': [1, 2, 7], 'pres': [7, -2, -2, -2, -2, -2, -2, -29, 224, 869, 36, 1262, 58, 1571, 80, 1684, 102, 87814, 43917]}, 'Perm': {'d': 93, 'gens': [13129506660262767975256285920361289487546915763638539005541811428509651849676564573962871719019539970174693978885815076199357420467650560000000, 25695708983488034985709191918611418790928177933411030898172468862434339058056550325843854792881993162308903443678109298342196036684336576681647, 38278253238027571513311501279299137905980555498423666332479864518503417860664942805103793594977134602002090622252024254198623611807334400000000, 50853370596870140557581529793858486361059448266330914160976141919296647110527719971541721843606999409678419181728174879251046380014141440000000, 63428486077576486827006712161749157886670709629355502194440063091073374422112780565653125659012612161268892518361879837611283231838044160000000, 76003603655065218098038664376065120279468997388548641397462217818030694609116198405280994291577682226947900499336796865605782372808130560000000, 326697157739096593111937363293]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{29}:Q_{64}', 'transitive_degree': 1856, 'wreath_data': None, 'wreath_product': False}
