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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '5280.z', 'ambient_counter': 26, 'ambient_order': 5280, 'ambient_tex': 'F_{11}\\times \\GL(2,3)', 'central': False, 'central_factor': False, 'centralizer_order': 220, 'characteristic': False, 'core_order': 2, 'counter': 178, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5280.z.660.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '660.e1.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': 660, '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': '8.3', 'subgroup_hash': 3, 'subgroup_order': 8, 'subgroup_tex': 'D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5280.z', 'aut_centralizer_order': None, 'aut_label': '660.e1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '24.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['60.e1.a1', '132.d1.a1', '330.a1.a1', '330.d1.a1', '330.e1.a1'], 'contains': ['1320.b1.a1', '1320.d1.a1'], 'core': '2640.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [3740, -1, 7256, -1, 3208, -1, 7261, -1], 'generators': [2751, 1320], 'label': '5280.z.660.e1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '110.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '660.e1.a1', 'projective_image': '2640.bv', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '660.e1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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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': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, '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': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [6, 10, 30, 20, 20], 'aut_gens': [[1, 2, 60, 120], [4041, 2702, 3960, 2580], [1121, 4642, 1380, 3240], [881, 5162, 4020, 3180], [3561, 5242, 1320, 2460], [5101, 622, 60, 900]], 'aut_group': None, 'aut_hash': 8877223747992235418, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10560, 'aut_permdeg': 34, 'aut_perms': [285854544689881142358057741809216128858, 237167178589744633213480103025514898537, 254670729274486509200506753210133121939, 205497630369727930667420271285611154195, 257303293177107038619787024216299808385], 'aut_phi_ratio': 8.25, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 2, 1], [2, 12, 1, 1], [2, 132, 1, 1], [3, 8, 1, 1], [4, 6, 1, 1], [4, 66, 1, 1], [5, 11, 1, 4], [6, 8, 1, 1], [6, 88, 2, 1], [8, 6, 2, 1], [8, 66, 2, 1], [10, 11, 1, 4], [10, 11, 2, 4], [10, 132, 1, 8], [11, 10, 1, 1], [15, 88, 1, 4], [20, 66, 1, 8], [22, 10, 1, 1], [22, 120, 1, 1], [30, 88, 1, 4], [30, 88, 2, 4], [33, 80, 1, 1], [40, 66, 2, 8], [44, 60, 1, 1], [66, 80, 1, 1], [88, 60, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_{11}\\times A_4).C_5.C_2^4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 660, 'autcentquo_group': '2640.bv', 'autcentquo_hash': 5844094576270439126, 'autcentquo_nilpotent': False, 'autcentquo_order': 2640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 11, 2], [2, 12, 1], [2, 132, 1], [3, 8, 1], [4, 6, 1], [4, 66, 1], [5, 11, 4], [6, 8, 1], [6, 88, 2], [8, 6, 2], [8, 66, 2], [10, 11, 12], [10, 132, 8], [11, 10, 1], [15, 88, 4], [20, 66, 8], [22, 10, 1], [22, 120, 1], [30, 88, 12], [33, 80, 1], [40, 66, 16], [44, 60, 1], [66, 80, 1], [88, 60, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '2640.bv', 'commutator_count': 1, 'commutator_label': '264.12', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1', '11.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 26, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['110.1', 1], ['48.29', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 1, 2], [2, 12, 1, 1], [2, 132, 1, 1], [3, 8, 1, 1], [4, 6, 1, 1], [4, 66, 1, 1], [5, 11, 4, 1], [6, 8, 1, 1], [6, 88, 1, 2], [8, 6, 2, 1], [8, 66, 2, 1], [10, 11, 4, 3], [10, 132, 4, 2], [11, 10, 1, 1], [15, 88, 4, 1], [20, 66, 4, 2], [22, 10, 1, 1], [22, 120, 1, 1], [30, 88, 4, 3], [33, 80, 1, 1], [40, 66, 8, 2], [44, 60, 1, 1], [66, 80, 1, 1], [88, 60, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 432, 'exponent': 1320, 'exponents_of_order': [5, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[20, 0, 2], [40, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '2640.bv', 'hash': 7098873280367891470, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 660, 'inner_gen_orders': [2, 30, 2, 22], 'inner_gens': [[1, 1402, 2700, 2580], [2741, 2, 1380, 780], [2641, 3962, 60, 2760], [2941, 2102, 2700, 120]], 'inner_hash': 5844094576270439126, 'inner_nilpotent': False, 'inner_order': 2640, 'inner_split': True, 'inner_tex': 'S_4\\times F_{11}', 'inner_used': [1, 2, 4], 'irrC_degree': 20, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 40, 'irrep_stats': [[1, 20], [2, 30], [3, 20], [4, 10], [10, 2], [20, 3], [30, 2], [40, 1]], 'label': '5280.z', 'linC_count': 40, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 8, 'linQ_dim': 14, 'linQ_dim_count': 8, 'linR_count': 24, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'F11*GL(2,3)', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 67, 'number_characteristic_subgroups': 25, 'number_conjugacy_classes': 88, 'number_divisions': 35, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 172, 'number_subgroup_classes': 204, 'number_subgroups': 4082, 'old_label': None, 'order': 5280, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 167], [3, 8], [4, 72], [5, 44], [6, 184], [8, 144], [10, 1188], [11, 10], [15, 352], [20, 528], [22, 130], [30, 1056], [33, 80], [40, 1056], [44, 60], [66, 80], [88, 120]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 2642, 60, 120], [1, 3962, 60, 2760]], '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': 4, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [3, 4], [4, 8], [8, 2], [10, 2], [12, 4], [16, 4], [20, 1], [30, 2], [40, 2]], 'representations': {'PC': {'code': '434647004884680575732805436688320753475730673194948386432485859905125679126574278031193507378200130009', 'gens': [1, 2, 5, 6], 'pres': [8, -2, -2, -3, -5, -2, 2, -2, -11, 22433, 41, 96098, 90, 108004, 27612, 13220, 1796, 123845, 18733, 36741, 2429, 2245, 141, 282246, 114254, 10102, 5630, 166, 307207, 92175, 23063, 12831]}, 'Perm': {'d': 19, 'gens': [83722845284039587, 83808467941692982, 64408451193890097, 33838978104743691]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'F_{11}\\times \\GL(2,3)', 'transitive_degree': 88, 'wreath_data': None, 'wreath_product': False}
