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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1120.866', 'ambient_counter': 866, 'ambient_order': 1120, 'ambient_tex': 'C_{140}:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 560, 'characteristic': False, 'core_order': 4, 'counter': 166, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1120.866.280.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '280.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '280.26', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 26, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 280, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{140}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1120.866', 'aut_centralizer_order': 26880, 'aut_label': '280.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '2.1', 'aut_weyl_index': 53760, 'centralizer': '2.b1.a1', 'complements': ['4.h1.a1', '4.h1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['40.b1.a1', '56.b1.a1', '140.a1.a1', '140.f1.a1', '140.f1.b1'], 'contains': ['560.a1.a1', '560.d1.a1'], 'core': '280.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1387, 8181, 1658, 3082, 1832, 1924, 2046, 3678], 'generators': [1, 4], 'label': '1120.866.280.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '280.b1.a1', 'normal_contained_in': ['40.b1.a1', '56.b1.a1', '140.a1.a1'], 'normal_contains': ['560.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '280.b1.a1', 'projective_image': '560.158', 'quotient_action_image': '2.1', 'quotient_action_kernel': '140.4', 'quotient_action_kernel_order': 140, 'quotient_fusion': None, 'short_label': '280.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': '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': 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, 3]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', '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]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [12, 30, 6, 6, 12, 12, 2, 4], 'aut_gens': [[1, 2, 8], [565, 282, 536], [5, 378, 1048], [565, 786, 872], [561, 770, 1032], [561, 822, 24], [1, 742, 940], [565, 362, 1112], [5, 618, 220]], 'aut_group': None, 'aut_hash': 8888889471875883849, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 107520, 'aut_permdeg': 284, 'aut_perms': 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'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 420, 'autcentquo_group': '1680.934', 'autcentquo_hash': 934, 'autcentquo_nilpotent': False, 'autcentquo_order': 1680, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{70}:C_{12}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 140, 2], [4, 2, 4], [4, 140, 2], [5, 2, 2], [7, 2, 3], [10, 2, 14], [14, 2, 21], [20, 2, 16], [28, 2, 24], [35, 2, 12], [70, 2, 84], [140, 2, 96]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '280.39', 'commutator_count': 1, 'commutator_label': '140.11', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 866, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 140, 1, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 140, 1, 2], [5, 2, 2, 1], [7, 2, 3, 1], [10, 2, 2, 3], [10, 2, 4, 2], [14, 2, 3, 3], [14, 2, 6, 2], [20, 2, 4, 2], [20, 2, 8, 1], [28, 2, 6, 2], [28, 2, 12, 1], [35, 2, 12, 1], [70, 2, 12, 3], [70, 2, 24, 2], [140, 2, 24, 2], [140, 2, 48, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4032, 'exponent': 140, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '280.39', 'hash': 866, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [2, 2, 70], 'inner_gens': [[1, 6, 8], [5, 2, 1112], [1, 18, 8]], 'inner_hash': 39, 'inner_nilpotent': False, 'inner_order': 280, 'inner_split': False, 'inner_tex': 'C_2\\times D_{70}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 278]], 'label': '1120.866', 'linC_count': 13824, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 64, 'linQ_dim': 14, 'linQ_dim_count': 64, 'linR_count': 96, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C140:D4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 39, 'number_conjugacy_classes': 286, 'number_divisions': 40, 'number_normal_subgroups': 63, 'number_subgroup_autclasses': 120, 'number_subgroup_classes': 188, 'number_subgroups': 2364, 'old_label': None, 'order': 1120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 287], [4, 288], [5, 4], [7, 6], [10, 28], [14, 42], [20, 32], [28, 48], [35, 24], [70, 168], [140, 192]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 840, 0, 0], 'outer_gens': [[561, 2, 8], [5, 2, 8], [1, 2, 12], [1, 282, 8], [5, 2, 568], [1, 2, 1096]], 'outer_group': '384.20153', 'outer_hash': 20153, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 17, 'outer_perms': [20922789888000, 3669120, 87178291200, 479001600, 40320, 41187], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\\times C_{12}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 5], [6, 4], [8, 4], [12, 4], [16, 1], [24, 5], [48, 4], [96, 1]], 'representations': {'PC': {'code': 467384232044391702931625224930546229015, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -2, -2, -2, -5, -7, 85, 36, 15578, 80, 19331, 102, 22860, 250, 23533]}, 'Perm': {'d': 20, 'gens': [6423384156542765, 1037836809, 83462400, 1037836800, 16, 50520, 134491780578355200]}}, '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': 'C_{140}:D_4', 'transitive_degree': 560, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [12, 6, 30, 6, 12], 'aut_gens': [[1, 2], [119, 46], [225, 38], [205, 122], [9, 78], [17, 174]], 'aut_group': None, 'aut_hash': 1492037419303921345, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6720, 'aut_permdeg': 140, 'aut_perms': [4106427697375651327316970805280107228296795813220345987516263211398172714643296998927000456141886254624919206876638733130473819432463118376878240676174166745361743480119749216380153346532495590184806286438248679402142516911823861373835188976, 4281219153971300929868069855169233041917094843740621106207786692581378643237469482129628884142822529514154649764077861625940230509711170213319147456408501070519196425604544255898332261842423197668459810813486246920906829018857804898835570154, 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'aut_supersolvable': True, 'aut_tex': 'C_{70}.(C_2^3\\times C_{12})', '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': 420, 'autcentquo_group': '1680.934', 'autcentquo_hash': 934, 'autcentquo_nilpotent': False, 'autcentquo_order': 1680, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{70}:C_{12}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 70, 2], [4, 2, 1], [5, 2, 2], [7, 2, 3], [10, 2, 2], [14, 2, 3], [20, 2, 4], [28, 2, 6], [35, 2, 12], [70, 2, 12], [140, 2, 24]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '140.10', 'commutator_count': 1, 'commutator_label': '70.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 26, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 70, 1, 2], [4, 2, 1, 1], [5, 2, 2, 1], [7, 2, 3, 1], [10, 2, 2, 1], [14, 2, 3, 1], [20, 2, 4, 1], [28, 2, 6, 1], [35, 2, 12, 1], [70, 2, 12, 1], [140, 2, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 140, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': [[2, 1, 24]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '140.10', 'hash': 26, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [2, 70], 'inner_gens': [[1, 278], [5, 2]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 140, 'inner_split': True, 'inner_tex': 'D_{70}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 69]], 'label': '280.26', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 4, 'linR_count': 24, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D140', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 13, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 73, 'number_divisions': 14, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 32, 'number_subgroups': 348, 'old_label': None, 'order': 280, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 141], [4, 2], [5, 4], [7, 6], [10, 4], [14, 6], [20, 8], [28, 12], [35, 24], [70, 24], [140, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [210, 0, 0], 'outer_gens': [[71, 2], [1, 142], [1, 146]], 'outer_group': '48.44', 'outer_hash': 44, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [3628800, 40320, 2163], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{12}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 2], [6, 2], [8, 1], [12, 1], [24, 2], [48, 1]], 'representations': {'PC': {'code': 312915836415372159230807, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -5, -7, 2781, 26, 4142, 42, 5443, 118, 6004]}, 'GLFp': {'d': 2, 'p': 139, 'gens': [65336835, 370615423]}, 'Perm': {'d': 16, 'gens': [87660962047, 7297920, 11652480, 1482509952000, 973]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{140}', 'transitive_degree': 140, 'wreath_data': None, 'wreath_product': False}
