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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '74112.g', 'ambient_counter': 7, 'ambient_order': 74112, 'ambient_tex': 'D_{193}:C_{192}', 'central': False, 'central_factor': False, 'centralizer_order': 384, 'characteristic': False, 'core_order': 6, 'counter': 44, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '74112.g.386.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '386.b1.b1', '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': 386, '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': '192.2', 'subgroup_hash': 2, 'subgroup_order': 192, 'subgroup_tex': 'C_{192}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '74112.g', 'aut_centralizer_order': None, 'aut_label': '386.b1', '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': ['2.b1.b1', '193.a1.a1'], 'contains': ['772.c1.a1', '1158.b1.b1'], 'core': '12352.a1.a1', 'coset_action_label': None, 'count': 193, 'diagramx': [971, -1, 1520, -1, 8697, -1, 8324, -1], 'generators': [99, 42320, 42264, 24704, 37056, 11718, 66060], 'label': '74112.g.386.b1.b1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '193.a1.a1', 'old_label': '386.b1.b1', 'projective_image': '12352.1683', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '386.b1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '192.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 16, 'aut_gen_orders': [2, 2, 16], 'aut_gens': [[1], [65], [127], [133]], 'aut_group': '64.183', 'aut_hash': 183, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 20, 'aut_perms': [355687428096000, 121652215820507520, 121648016789608890], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [8, 1, 4, 1], [12, 1, 4, 1], [16, 1, 8, 1], [24, 1, 8, 1], [32, 1, 16, 1], [48, 1, 16, 1], [64, 1, 32, 1], [96, 1, 32, 1], [192, 1, 64, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_{16}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '64.183', 'autcent_hash': 183, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{16}', '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], [3, 1, 2], [4, 1, 2], [6, 1, 2], [8, 1, 4], [12, 1, 4], [16, 1, 8], [24, 1, 8], [32, 1, 16], [48, 1, 16], [64, 1, 32], [96, 1, 32], [192, 1, 64]], 'center_label': '192.2', 'center_order': 192, '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', '2.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['64.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [8, 1, 4, 1], [12, 1, 4, 1], [16, 1, 8, 1], [24, 1, 8, 1], [32, 1, 16, 1], [48, 1, 16, 1], [64, 1, 32, 1], [96, 1, 32, 1], [192, 1, 64, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 192, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 64]], 'familial': True, 'frattini_label': '32.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 64, 'irrQ_dim': 64, 'irrR_degree': 2, 'irrep_stats': [[1, 192]], 'label': '192.2', 'linC_count': 64, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 34, 'linQ_degree_count': 2, 'linQ_dim': 34, 'linQ_dim_count': 2, 'linR_count': 32, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C192', 'ngens': 7, 'nilpotency_class': 1, 'nilpotent': True, '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': 14, 'number_conjugacy_classes': 192, 'number_divisions': 14, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 14, 'number_subgroups': 14, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [8, 4], [12, 4], [16, 8], [24, 8], [32, 16], [48, 16], [64, 32], [96, 32], [192, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [2, 2, 16], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[65], [127], [133]], 'outer_group': '64.183', 'outer_hash': 183, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 20, 'outer_perms': [355687428096000, 121652215820507520, 121648016789608890], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{16}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 67, 'pgroup': 0, 'primary_abelian_invariants': [64, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 2], [8, 2], [16, 2], [32, 2], [64, 1]], 'representations': {'PC': {'code': 4553336054269763133313, 'gens': [1], 'pres': [7, -2, -2, -2, -2, -2, -2, -3, 14, 36, 58, 80, 102, 124]}, 'GLFp': {'d': 2, 'p': 31, 'gens': [119074]}, 'Perm': {'d': 67, 'gens': [34812819462234954428607685341897082192301945708552738688865836491528209085355972467097600000000, 4, 17125731743961598921943674325579392227242124480672411041416803851981460023914660624295657472000, 8282187884824921168611668818381987555972846584752266182758164987540677113354309924032688179200, 3860415955256618126428729875983775624818504154288173718968029842331521501914651066812729456320, 1649530213841581475001430175259373643701598531447003215154104016101461640635225996085127825760, 544471857487925546941072384503952816534977960565454352280865841524404515036981945541028258664]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [192], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{192}', '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': False, 'abelian_quotient': '384.563', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 37056, 'aut_gen_orders': [16, 192, 192, 192, 48], 'aut_gens': [[1, 32], [42049, 20576], [49089, 35872], [42529, 2336], [20769, 35872], [14721, 38944]], 'aut_group': None, 'aut_hash': 2047870114700848400, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 592896, 'aut_permdeg': 774, 'aut_perms': 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'aut_phi_ratio': 24.125, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 193, 2, 1], [3, 1, 2, 1], [4, 1, 2, 1], [4, 193, 1, 2], [6, 1, 2, 1], [6, 193, 4, 1], [8, 193, 1, 4], [8, 193, 2, 2], [12, 1, 4, 1], [12, 193, 2, 2], [16, 193, 1, 8], [16, 193, 2, 4], [24, 193, 2, 4], [24, 193, 4, 2], [32, 193, 2, 16], [48, 193, 2, 8], [48, 193, 4, 4], [64, 193, 4, 16], [96, 193, 4, 16], [192, 193, 8, 16], [193, 32, 6, 1], [386, 32, 6, 1], [579, 32, 12, 1], [772, 32, 12, 1], [1158, 32, 12, 1], [2316, 32, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{386}.C_{96}.C_2^4', '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': 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], [3, 1, 2], [4, 1, 2], [4, 193, 2], [6, 1, 2], [6, 193, 4], [8, 193, 8], [12, 1, 4], [12, 193, 4], [16, 193, 16], [24, 193, 16], [32, 193, 32], [48, 193, 32], [64, 193, 64], [96, 193, 64], [192, 193, 128], [193, 32, 6], [386, 32, 6], [579, 32, 12], [772, 32, 12], [1158, 32, 12], [2316, 32, 24]], 'center_label': '12.2', 'center_order': 12, 'central_product': True, 'central_quotient': '6176.247', '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', '2.1', '3.1', '193.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['24704.21814', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 193, 1, 2], [3, 1, 2, 1], [4, 1, 2, 1], [4, 193, 2, 1], [6, 1, 2, 1], [6, 193, 2, 2], [8, 193, 4, 2], [12, 1, 4, 1], [12, 193, 4, 1], [16, 193, 8, 2], [24, 193, 8, 2], [32, 193, 16, 2], [48, 193, 16, 2], [64, 193, 32, 2], [96, 193, 32, 2], [192, 193, 64, 2], [193, 32, 6, 1], [386, 32, 6, 1], [579, 32, 12, 1], [772, 32, 12, 1], [1158, 32, 12, 1], [2316, 32, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3072, 'exponent': 37056, 'exponents_of_order': [7, 1, 1], 'factors_of_aut_order': [2, 3, 193], 'factors_of_order': [2, 3, 193], 'faithful_reps': [[32, 0, 24]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '37056.r', 'hash': 6586773659675919904, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6176, 'inner_gen_orders': [32, 193], 'inner_gens': [[1, 6944], [67201, 32]], 'inner_hash': 247, 'inner_nilpotent': False, 'inner_order': 6176, 'inner_split': False, 'inner_tex': 'C_{193}:C_{32}', 'inner_used': [1, 2], 'irrC_degree': 32, 'irrQ_degree': 768, 'irrQ_dim': 768, 'irrR_degree': 64, 'irrep_stats': [[1, 384], [32, 72]], 'label': '74112.g', 'linC_count': 24, 'linC_degree': 32, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 226, 'linQ_degree_count': 16, 'linQ_dim': 226, 'linQ_dim_count': 16, 'linR_count': 768, 'linR_degree': 34, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D193:C192', 'ngens': 9, '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 118, 'number_characteristic_subgroups': 38, 'number_conjugacy_classes': 456, 'number_divisions': 34, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 80, 'number_subgroups': 6608, 'old_label': None, 'order': 74112, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 387], [3, 2], [4, 388], [6, 774], [8, 1544], [12, 776], [16, 3088], [24, 3088], [32, 6176], [48, 6176], [64, 12352], [96, 12352], [192, 24704], [193, 192], [386, 192], [579, 384], [772, 384], [1158, 384], [2316, 768]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [0, 5, 5, 61782], 'outer_gens': [[24705, 37024], [61761, 39328], [6177, 59488], [6177, 25888]], 'outer_group': '96.221', 'outer_hash': 221, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [2424, 1680, 40320, 11612164], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:C_2^3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 260, 'pgroup': 0, 'primary_abelian_invariants': [2, 64, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 4], [8, 4], [16, 4], [32, 4], [64, 2], [192, 2], [384, 3], [768, 1]], 'representations': {'PC': {'code': '365644094072947909894236552328535569515916202899457241062082746986993595775394582645118793487688647782911948228437052420822502342088632839277002784515931219220230046079', 'gens': [1, 6], 'pres': [9, -2, -2, -2, -2, -2, -2, -2, -3, -193, 18, 46, 74, 102, 34780, 374981, 664430, 337415, 184280, 41621, 158, 874950, 1550319, 787272, 429945, 97062, 186, 1999879, 875536, 465433, 315682, 221803, 286, 746504, 2954897, 70010, 314963, 373292]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [7189251, 1006468143, 64701556]}, 'Perm': {'d': 260, 'gens': 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