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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '254016.a', 'ambient_counter': 1, 'ambient_order': 254016, 'ambient_tex': '\\SL(2,8)^2', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 25, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '254016.a.1134.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '1134.a1', 'outer_equivalence': True, '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': 1134, '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': '224.195', 'subgroup_hash': 195, 'subgroup_order': 224, 'subgroup_tex': 'C_2^2\\times F_8', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '254016.a', 'aut_centralizer_order': None, 'aut_label': '1134.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '31752.e1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['126.a1', '567.a1'], 'contains': ['2268.a1', '7938.b1', '9072.b1'], 'core': '254016.a1', 'coset_action_label': None, 'count': 1134, 'diagramx': None, 'generators': [21115907366400, 64430988156298, 129903397747200, 86418678124800, 148048896701695, 49966656364800], 'label': '254016.a.1134.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '567.a1', 'old_label': '1134.a1', 'projective_image': '254016.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1134.a1', 'subgroup_fusion': None, 'weyl_group': '56.11'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '28.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 42, 'aut_gen_orders': [2, 3, 3, 7, 2, 2, 2], 'aut_gens': [[7, 16, 4083840, 40723320, 90842400, 135732240], [16, 7, 4083840, 40723320, 90842400, 135732240], [7, 16, 8508240, 303182760, 262459440, 90842400], [16, 23, 4083840, 40723320, 90842400, 135732240], [7, 16, 4083840, 135732240, 212340360, 303182760], [7, 16, 221729760, 40723320, 90842400, 135732240], [7, 16, 251193720, 40723320, 90842400, 135732240], [7, 16, 81453000, 40723320, 90842400, 135732240]], 'aut_group': '1008.885', 'aut_hash': 885, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1008, 'aut_permdeg': 11, 'aut_perms': [1, 949080, 3, 867000, 3719544, 19441464, 15301704], 'aut_phi_ratio': 10.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 7, 1, 1], [2, 7, 3, 1], [7, 8, 3, 2], [14, 8, 9, 2]], 'aut_supersolvable': False, 'aut_tex': 'F_8:C_3\\times 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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.43', 'autcentquo_hash': 43, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_8:C_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 7, 4], [7, 8, 6], [14, 8, 18]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '56.11', 'commutator_count': 1, 'commutator_label': '8.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 195, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['56.11', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 7, 1, 4], [7, 8, 6, 1], [14, 8, 6, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 16, 'exponent': 14, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '224.195', 'hash': 195, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [1, 1, 7, 2, 2, 2], 'inner_gens': [[7, 16, 4083840, 40723320, 90842400, 135732240], [7, 16, 4083840, 40723320, 90842400, 135732240], [7, 16, 4083840, 90842400, 135732240, 212340360], [7, 16, 165288360, 40723320, 90842400, 135732240], [7, 16, 251193720, 40723320, 90842400, 135732240], [7, 16, 51989040, 40723320, 90842400, 135732240]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': True, 'inner_tex': 'F_8', 'inner_used': [3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 28], [7, 4]], 'label': '224.195', 'linC_count': 42, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 6, 'linQ_dim': 8, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*F8', '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': 8, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 32, 'number_divisions': 12, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 72, 'number_subgroups': 419, 'old_label': None, 'order': 224, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 31], [7, 48], [14, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[23, 16, 8508240, 135732240, 303182760, 262459440], [16, 23, 4083840, 40723320, 90842400, 135732240]], 'outer_group': '18.3', 'outer_hash': 3, 'outer_nilpotent': False, 'outer_order': 18, 'outer_permdeg': 6, 'outer_perms': [305, 466], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3\\times S_3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [6, 4], [7, 4]], 'representations': {'PC': {'code': 18651277966320690975961272, 'gens': [1, 2, 4, 5, 6], 'pres': [6, -2, -2, -7, -2, 2, 2, 31, 2361, 687, 2530, 646, 2531, 521]}, 'GLFq': {'d': 3, 'q': 8, 'gens': [101276545, 16849409, 16880641, 85079488, 16784385, 16979969]}, 'Perm': {'d': 12, 'gens': [7, 16, 4083840, 40723320, 90842400, 135732240]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 14], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times F_8', 'transitive_degree': 28, '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': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 252, 'aut_gen_orders': [12, 6], 'aut_gens': [[379412420931487, 356995585140781], [1821876841513146, 1131399943319392], [112735425294193, 1535390371649655]], 'aut_group': '4572288.a', 'aut_hash': 6567295075771123923, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4572288, 'aut_permdeg': 112, 'aut_perms': [177972985008229000669429311816417853173541465557431509964071434549398519179740150859151312528501502595813214063585382299046221361185366450356568211442292320240915649227316053411014249, 192081091051585489545847521005098043379839540284921027544507549558185448443593532048292589613500776468688314293319114826158290818243409578060797132310850306815574000894638962406541922], 'aut_phi_ratio': 63.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 63, 2, 1], [2, 3969, 1, 1], [3, 56, 2, 1], [3, 3136, 1, 1], [6, 3528, 2, 1], [7, 72, 6, 1], [7, 5184, 9, 1], [9, 56, 6, 1], [9, 3136, 6, 1], [9, 3136, 9, 1], [14, 4536, 6, 1], [18, 3528, 6, 1], [21, 4032, 6, 1], [63, 4032, 18, 1]], 'aut_supersolvable': False, 'aut_tex': '{}^2G(2,3)\\wr C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 252, 'autcentquo_group': '4572288.a', 'autcentquo_hash': 6567295075771123923, 'autcentquo_nilpotent': False, 'autcentquo_order': 4572288, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '{}^2G(2,3)\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 63, 2], [2, 3969, 1], [3, 56, 2], [3, 3136, 1], [6, 3528, 2], [7, 72, 6], [7, 5184, 9], [9, 56, 6], [9, 3136, 15], [14, 4536, 6], [18, 3528, 6], [21, 4032, 6], [63, 4032, 18]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '254016.a', 'commutator_count': 1, 'commutator_label': '254016.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['504.156', '504.156'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [['504.156', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 63, 1, 2], [2, 3969, 1, 1], [3, 56, 1, 2], [3, 3136, 1, 1], [6, 3528, 1, 2], [7, 72, 3, 2], [7, 5184, 3, 3], [9, 56, 3, 2], [9, 3136, 3, 5], [14, 4536, 3, 2], [18, 3528, 3, 2], [21, 4032, 3, 2], [63, 4032, 9, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 10011, 'exponent': 126, 'exponents_of_order': [6, 4, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[49, 1, 16], [56, 1, 8], [63, 1, 24], [64, 1, 1], [72, 1, 6], [81, 1, 9]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '254016.a', 'hash': 704524454924516110, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 126, 'inner_gen_orders': [18, 18], 'inner_gens': [[379412420931487, 2533420861335052], [1072481229682436, 356995585140781]], 'inner_hash': 704524454924516110, 'inner_nilpotent': False, 'inner_order': 254016, 'inner_split': True, 'inner_tex': '\\SL(2,8)^2', 'inner_used': [1, 2], 'irrC_degree': 49, 'irrQ_degree': 49, 'irrQ_dim': 49, 'irrR_degree': 49, 'irrep_stats': [[1, 1], [7, 8], [8, 2], [9, 6], [49, 16], [56, 8], [63, 24], [64, 1], [72, 6], [81, 9]], 'label': '254016.a', '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': False, 'metacyclic': False, 'monomial': False, 'name': 'SL(2,8)^2', 'ngens': 2, '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, 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, 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, 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': 2, 'number_conjugacy_classes': 81, 'number_divisions': 29, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 122, 'number_subgroup_classes': 276, 'number_subgroups': 682507, 'old_label': None, 'order': 254016, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 4095], [3, 3248], [6, 7056], [7, 47088], [9, 47376], [14, 27216], [18, 21168], [21, 24192], [63, 72576]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [6, 3], 'outer_gen_pows': [1137677701921465, 356056299263167], 'outer_gens': [[1110217059184057, 1196629346725567], [418818002988000, 153460181322762]], 'outer_group': '18.3', 'outer_hash': 3, 'outer_nilpotent': False, 'outer_order': 18, 'outer_permdeg': 6, 'outer_perms': [498, 348], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3\\times S_3', 'pc_rank': None, 'perfect': True, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [7, 2], [8, 2], [21, 2], [27, 2], [49, 1], [56, 2], [64, 1], [147, 5], [168, 2], [189, 2], [216, 2], [243, 3], [567, 2]], 'representations': {'Lie': [{'d': 4, 'q': 8, 'gens': [141326051967552, 145144133587464], 'family': 'OmegaPlus'}, {'d': 4, 'q': 8, 'family': 'POmegaPlus'}, {'d': 4, 'q': 8, 'gens': [43981538885633, 35219805601793, 35185445871617, 52777631907841, 39583492374529, 35185445863441, 35185445863433, 35185445867521, 35185445863457, 35202625732609, 35194035798017, 35185445879809], 'family': 'SpinPlus'}], 'Perm': {'d': 18, 'gens': [379412420931487, 356995585140781]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SL(2,8)^2', 'transitive_degree': 81, 'wreath_data': None, 'wreath_product': False}
