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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '4400.j', 'ambient_counter': 10, 'ambient_order': 4400, 'ambient_tex': 'C_{44}:C_{10}^2', 'central': False, 'central_factor': False, 'centralizer_order': 220, 'characteristic': False, 'core_order': 55, 'counter': 53, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '4400.j.40.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '40.e1', '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': 40, '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': '110.6', 'subgroup_hash': 6, 'subgroup_order': 110, 'subgroup_tex': 'C_{110}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4400.j', 'aut_centralizer_order': 220, 'aut_label': '40.e1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '40.9', 'aut_weyl_index': 880, 'centralizer': '20.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['8.c1', '20.a1', '20.k1'], 'contains': ['80.a1', '200.c1', '440.c1'], 'core': '80.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [8388, -1, 1438, -1], 'generators': [50, 400, 882], 'label': '4400.j.40.e1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '20.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1', 'old_label': '40.e1', 'projective_image': '880.118', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '40.e1', 'subgroup_fusion': None, 'weyl_group': '10.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '110.6', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 20, 'aut_gen_orders': [2, 20], 'aut_gens': [[1], [21], [97]], 'aut_group': '40.9', 'aut_hash': 9, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 40, 'aut_permdeg': 11, 'aut_perms': [3719520, 46113], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1], [11, 1, 10, 1], [22, 1, 10, 1], [55, 1, 40, 1], [110, 1, 40, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{20}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '40.9', 'autcent_hash': 9, 'autcent_nilpotent': True, 'autcent_order': 40, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{20}', '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], [5, 1, 4], [10, 1, 4], [11, 1, 10], [22, 1, 10], [55, 1, 40], [110, 1, 40]], 'center_label': '110.6', 'center_order': 110, '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', '5.1', '11.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['2.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1], [11, 1, 10, 1], [22, 1, 10, 1], [55, 1, 40, 1], [110, 1, 40, 1]], 'element_repr_type': 'PC', 'elementary': 110, 'eulerian_function': 1, 'exponent': 110, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[1, 0, 40]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '110.6', 'hash': 6, 'hyperelementary': 110, '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': 40, 'irrQ_dim': 40, 'irrR_degree': 2, 'irrep_stats': [[1, 110]], 'label': '110.6', 'linC_count': 40, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 3, 'linQ_dim': 14, 'linQ_dim_count': 3, 'linR_count': 20, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C110', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_conjugacy_classes': 110, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 110, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [5, 4], [10, 4], [11, 10], [22, 10], [55, 40], [110, 40]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [2, 20], 'outer_gen_pows': [0, 0], 'outer_gens': [[21], [97]], 'outer_group': '40.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 40, 'outer_permdeg': 11, 'outer_perms': [3719520, 46113], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{20}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 5, 11], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [10, 2], [40, 2]], 'representations': {'PC': {'code': 3875039, 'gens': [1], 'pres': [3, -2, -5, -11, 6, 34]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [1343, 13320, 2664]}, 'Perm': {'d': 18, 'gens': [355687428096000, 5230697472000, 36288000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [110], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{110}', 'transitive_degree': 110, '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': '200.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [20, 20, 20, 20, 20, 10], 'aut_gens': [[1, 10, 100], [2583, 756, 300], [1501, 3574, 300], [4165, 3774, 4100], [1, 1838, 2500], [101, 1938, 1500], [1601, 2610, 3100]], 'aut_group': None, 'aut_hash': 2437826575940019868, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 35200, 'aut_permdeg': 81, 'aut_perms': [4555388998179032647089305550864848335304769513432943178265188568622987650625154040138747736440211584547246988808352492955, 3020275665081616031422391807499702152788409405642771204441617566138958051699474777274271653767785642036586948012823366147, 678615547409289874301823831385976391706417033364048070217212857434801433985228892083822953000902987656219551668994150335, 5496778742352979675825693466198549807708382509728876385776560889902546603416618919712330836826510531825169096589037693984, 2527705418138644378424441807050802391714847636576206926482056116327465501024334099191350906123287611713887560887835566512, 3276114896137516766636654758155380959030296635384240992820167249871469644618553785722331608137995498427076016910649595491], 'aut_phi_ratio': 22.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [2, 11, 2, 1], [2, 22, 2, 1], [4, 2, 1, 1], [4, 22, 1, 1], [5, 1, 4, 1], [5, 11, 5, 4], [10, 1, 4, 1], [10, 2, 8, 1], [10, 11, 5, 4], [10, 11, 8, 1], [10, 11, 10, 4], [10, 22, 8, 1], [10, 22, 10, 8], [11, 10, 1, 1], [20, 2, 4, 1], [20, 22, 4, 1], [20, 22, 5, 8], [22, 10, 1, 1], [22, 20, 2, 1], [44, 20, 1, 1], [55, 10, 4, 1], [110, 10, 4, 1], [110, 20, 8, 1], [220, 20, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{110}.C_{10}.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '160.236', 'autcent_hash': 236, 'autcent_nilpotent': False, 'autcent_order': 160, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times F_5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '220.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 220, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [2, 11, 2], [2, 22, 2], [4, 2, 1], [4, 22, 1], [5, 1, 4], [5, 11, 20], [10, 1, 4], [10, 2, 8], [10, 11, 68], [10, 22, 88], [11, 10, 1], [20, 2, 4], [20, 22, 44], [22, 10, 1], [22, 20, 2], [44, 20, 1], [55, 10, 4], [110, 10, 4], [110, 20, 8], [220, 20, 4]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '440.42', 'commutator_count': 1, 'commutator_label': '22.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['110.1', 1], ['5.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [2, 11, 1, 2], [2, 22, 1, 2], [4, 2, 1, 1], [4, 22, 1, 1], [5, 1, 4, 1], [5, 11, 4, 5], [10, 1, 4, 1], [10, 2, 4, 2], [10, 11, 4, 17], [10, 22, 4, 22], [11, 10, 1, 1], [20, 2, 4, 1], [20, 22, 4, 11], [22, 10, 1, 1], [22, 20, 1, 2], [44, 20, 1, 1], [55, 10, 4, 1], [110, 10, 4, 1], [110, 20, 4, 2], [220, 20, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 749952, 'exponent': 220, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[20, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '2200.a', 'hash': 532271021746972557, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 10, 22], 'inner_gens': [[1, 1610, 1900], [2801, 10, 300], [2601, 4210, 100]], 'inner_hash': 42, 'inner_nilpotent': False, 'inner_order': 440, 'inner_split': True, 'inner_tex': 'C_2^2\\times F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': 20, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 40, 'irrep_stats': [[1, 200], [2, 50], [10, 20], [20, 5]], 'label': '4400.j', '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': 'C44:C10^2', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 50, 'number_characteristic_subgroups': 35, 'number_conjugacy_classes': 275, 'number_divisions': 80, 'number_normal_subgroups': 164, 'number_subgroup_autclasses': 136, 'number_subgroup_classes': 432, 'number_subgroups': 3160, 'old_label': None, 'order': 4400, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 71], [4, 24], [5, 224], [10, 2704], [11, 10], [20, 976], [22, 50], [44, 20], [55, 40], [110, 200], [220, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 20, 'outer_gen_orders': [2, 4, 10], 'outer_gen_pows': [0, 0, 1564], 'outer_gens': [[3801, 10, 2100], [2065, 2492, 2100], [3165, 2374, 2100]], 'outer_group': '80.50', 'outer_hash': 50, 'outer_nilpotent': False, 'outer_order': 80, 'outer_permdeg': 9, 'outer_perms': [7, 5896, 122543], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times F_5', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 48], [8, 12], [10, 4], [20, 1], [40, 4], [80, 1]], 'representations': {'PC': {'code': '18053028909108461072444892686717030496013806886236833688186436148639481314671890303009', 'gens': [1, 3, 5], 'pres': [7, -2, -5, -2, -5, -2, -2, -11, 14, 33812, 12714, 58, 56563, 5890, 66504, 15761, 1068, 1600, 102, 159605, 37812, 2539, 3806, 124, 156806, 88213, 5900, 8847]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 31471887339564514, 10885681761182519, 10522401710269773, 19623198237756269, 33418192856010432, 11184584018506052]}, 'Perm': {'d': 20, 'gens': [34, 13606353239673671, 142700251934419991, 244009609488732240, 41074, 271082647280885113, 33455996122389911]}}, 'schur_multiplier': [2, 2, 10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{44}:C_{10}^2', 'transitive_degree': 220, 'wreath_data': None, 'wreath_product': False}
