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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '7776.is', 'ambient_counter': 227, 'ambient_order': 7776, 'ambient_tex': 'C_3^4:(D_4\\times D_6)', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 9, 'counter': 357, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '7776.is.54.r1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '54.r1', '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': 54, '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': '144.144', 'subgroup_hash': 144, 'subgroup_order': 144, 'subgroup_tex': 'S_3\\times D_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7776.is', 'aut_centralizer_order': None, 'aut_label': '54.r1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '3888.d1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['6.j1', '18.y1', '27.a1'], 'contains': ['108.bz1', '108.ca1', '108.cv1', '108.cx1', '108.cy1', '162.o1', '162.q1'], 'core': '864.a1', 'coset_action_label': None, 'count': 54, 'diagramx': [9469, -1, 8815, -1], 'generators': [1037877120, 46638149552, 280778480640, 699550311600, 518, 199348214021], 'label': '7776.is.54.r1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.a1', 'old_label': '54.r1', 'projective_image': '7776.is', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '54.r1', 'subgroup_fusion': None, 'weyl_group': '144.192'}
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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': 12, 'aut_gen_orders': [2, 2, 2, 6, 12], 'aut_gens': [[1, 2, 12], [1, 82, 12], [73, 74, 12], [1, 74, 84], [5, 74, 60], [1, 134, 12]], 'aut_group': '576.8553', 'aut_hash': 8553, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 576, 'aut_permdeg': 12, 'aut_perms': [41091120, 127, 120, 168058800, 95074854], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [2, 6, 2, 1], [2, 18, 2, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 2, 1, 1], [4, 6, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 2, 1], [6, 12, 2, 1], [12, 2, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [12, 6, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4:D_6^2', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '72.46', 'autcentquo_hash': 46, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [2, 6, 2], [2, 18, 2], [3, 2, 2], [3, 4, 1], [4, 2, 1], [4, 6, 1], [6, 2, 2], [6, 4, 1], [6, 6, 2], [6, 12, 2], [12, 2, 2], [12, 4, 3], [12, 6, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '72.46', 'commutator_count': 1, 'commutator_label': '18.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 144, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['24.6', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 1, 2], [2, 18, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [4, 2, 1, 1], [4, 6, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 1, 2], [6, 12, 1, 2], [12, 2, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [12, 6, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 12, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '72.46', 'hash': 144, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 6], 'inner_gens': [[1, 10, 12], [5, 2, 132], [1, 26, 12]], 'inner_hash': 46, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'S_3\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 14], [4, 5]], 'label': '144.144', 'linC_count': 18, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 48, 'linQ_dim': 6, 'linQ_dim_count': 48, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*D12', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 19, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 27, 'number_divisions': 24, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 74, 'number_subgroup_classes': 116, 'number_subgroups': 476, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 55], [3, 8], [4, 8], [6, 44], [12, 28]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [36, 0, 0], 'outer_gens': [[1, 38, 12], [1, 82, 60], [73, 2, 60]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 5], [8, 1]], 'representations': {'PC': {'code': 87380135466841110617737, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -3, -2, -2, -3, 121, 31, 146, 1593, 69, 1810, 88, 1739]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729070035819, 125124626797543173, 58301505185097352]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [99161, 59563, 92307, 7993, 5903, 5995]}, 'Perm': {'d': 10, 'gens': [1, 85704, 408240, 806400, 144, 3]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times D_{12}', 'transitive_degree': 24, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [12, 12, 6, 4, 6], 'aut_gens': [[199348214022, 295239213777, 18721024762, 99675923960], [274467957768, 301945190703, 699550312102, 476142877208], [301307650652, 635799513534, 442601188296, 469436890100], [301307650902, 723017756813, 442601187979, 481890850579], [281173944332, 481890850863, 389044016587, 301945190900], [636318507547, 113124292301, 275030381006, 711521682680]], 'aut_group': None, 'aut_hash': 9011329461466765991, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 31104, 'aut_permdeg': 30, 'aut_perms': [261966733110510185217203719038679, 101877027352541907850689779952553, 2255394837631398014925357525847, 235913559650099556389616568113811, 213395267144920269267785767296271], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 6, 2, 1], [2, 9, 1, 2], [2, 9, 2, 1], [2, 54, 2, 1], [2, 54, 4, 1], [2, 81, 1, 1], [2, 81, 2, 1], [3, 2, 1, 1], [3, 4, 2, 1], [3, 6, 2, 1], [3, 8, 2, 1], [3, 12, 1, 1], [3, 24, 4, 1], [3, 48, 2, 1], [4, 18, 1, 1], [4, 162, 1, 1], [4, 162, 2, 1], [6, 12, 2, 2], [6, 18, 1, 2], [6, 18, 2, 1], [6, 24, 2, 1], [6, 36, 2, 1], [6, 36, 4, 2], [6, 54, 2, 1], [6, 72, 2, 2], [6, 72, 4, 2], [6, 108, 1, 1], [6, 108, 2, 2], [6, 108, 4, 2], [6, 144, 2, 1], [6, 162, 1, 1], [6, 162, 2, 1], [6, 216, 2, 1], [6, 216, 4, 1], [12, 36, 1, 1], [12, 108, 2, 1], [12, 216, 1, 1], [12, 324, 1, 1], [12, 324, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_3:S_3\\times \\He_3).D_4^2', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 9011329461466765991, 'autcentquo_nilpotent': False, 'autcentquo_order': 31104, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(C_3:S_3\\times \\He_3).D_4^2', 'cc_stats': [[1, 1, 1], [2, 6, 2], [2, 9, 4], [2, 54, 6], [2, 81, 3], [3, 2, 1], [3, 4, 2], [3, 6, 2], [3, 8, 2], [3, 12, 1], [3, 24, 4], [3, 48, 2], [4, 18, 1], [4, 162, 3], [6, 12, 4], [6, 18, 4], [6, 24, 2], [6, 36, 10], [6, 54, 2], [6, 72, 12], [6, 108, 13], [6, 144, 2], [6, 162, 3], [6, 216, 6], [12, 36, 1], [12, 108, 2], [12, 216, 1], [12, 324, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '7776.is', 'commutator_count': 1, 'commutator_label': '486.231', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 227, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['108.17', 1], ['72.40', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 6, 1, 2], [2, 9, 1, 4], [2, 54, 1, 6], [2, 81, 1, 3], [3, 2, 1, 1], [3, 4, 1, 2], [3, 6, 1, 2], [3, 8, 1, 2], [3, 12, 1, 1], [3, 24, 1, 4], [3, 48, 1, 2], [4, 18, 1, 1], [4, 162, 1, 3], [6, 12, 1, 4], [6, 18, 1, 4], [6, 24, 1, 2], [6, 36, 1, 10], [6, 54, 1, 2], [6, 72, 1, 12], [6, 108, 1, 13], [6, 144, 1, 2], [6, 162, 1, 3], [6, 216, 1, 6], [12, 36, 1, 1], [12, 108, 1, 2], [12, 216, 1, 1], [12, 324, 1, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 33484389120, 'exponent': 12, 'exponents_of_order': [5, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 8]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '2592.fs', 'hash': 4994213685505676867, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 12, 6, 6], 'inner_gens': [[199348214022, 624303555581, 442601188364, 175793667859], [536170804488, 295239213777, 7225021964, 624303555139], [87660962587, 307693290717, 18721024762, 350193760760], [87660962343, 723017756851, 268324403962, 99675923960]], 'inner_hash': 4994213685505676867, 'inner_nilpotent': False, 'inner_order': 7776, 'inner_split': False, 'inner_tex': 'C_3^4:(D_4\\times D_6)', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 16], [2, 20], [4, 24], [6, 8], [8, 17], [12, 2], [16, 4], [24, 8]], 'label': '7776.is', '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': None, 'name': 'C3^4:(D4*D6)', 'ngens': 4, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 48, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 99, 'number_divisions': 99, 'number_normal_subgroups': 139, 'number_subgroup_autclasses': 1236, 'number_subgroup_classes': 3456, 'number_subgroups': 136108, 'old_label': None, 'order': 7776, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 615], [3, 242], [4, 504], [6, 4974], [12, 1440]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [82, 0], 'outer_gens': [[199348214204, 295239213655, 18721025227, 99675924260], [268719983622, 200262631377, 699550311802, 175793668040]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 24], [6, 8], [8, 17], [12, 2], [16, 4], [24, 8]], 'representations': {'PC': {'code': '1253194777663739017244900830872040030495784258809935097187894156093236383239509412124613077276334685871163599905611101502816810282151', 'gens': [1, 2, 4, 6, 9, 10], 'pres': [10, -2, -2, -3, -2, -3, -2, -2, -3, -3, 3, 31680, 63561, 51, 26162, 232803, 136573, 63503, 113, 146404, 58814, 7524, 211685, 75615, 39985, 6875, 8865, 175, 191526, 115936, 17666, 31116, 8026, 206, 23057, 51878, 4378, 1148, 302419, 8459, 3669]}, 'Perm': {'d': 15, 'gens': [199348214022, 295239213777, 18721024762, 99675923960]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:(D_4\\times D_6)', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
