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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '128.2216', 'ambient_counter': 2216, 'ambient_order': 128, 'ambient_tex': 'D_4^2:C_2', 'central': False, 'central_factor': False, 'centralizer_order': 64, 'characteristic': False, 'core_order': 1, 'counter': 141, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '128.2216.64.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '64.d1', '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': 64, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '128.2216', 'aut_centralizer_order': 1024, 'aut_label': '64.d1', 'aut_quo_index': None, 'aut_stab_index': 16, 'aut_weyl_group': '1.1', 'aut_weyl_index': 16384, 'centralizer': '2.c1', 'complements': None, 'conjugacy_class_count': 8, 'contained_in': ['32.b1', '32.l1', '32.q1', '32.s1', '32.t1'], 'contains': ['128.a1'], 'core': '128.a1', 'coset_action_label': None, 'count': 16, 'diagramx': [7285, -1, 7234, -1], 'generators': [212340360], 'label': '128.2216.64.d1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '32.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.c1', 'old_label': '64.d1', 'projective_image': '128.2216', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '64.d1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[11370967, 40723336, 90842405, 11291176, 135732256, 90842400, 16], [127463, 212340360, 5, 11291176, 135732256, 90842400, 16], [83463983, 167450536, 90842405, 11291176, 135732256, 90842400, 16], [40805407, 212340360, 262459454, 258829920, 135732256, 90842400, 90842416], [83463983, 167450520, 14, 11291160, 262459456, 90842400, 16], [83463967, 40723336, 90842405, 11291176, 135732256, 90842400, 16], [219718823, 40723336, 5, 143354280, 135732256, 90842400, 16], [83463967, 167450520, 90842405, 11291176, 135732256, 90842400, 16], [83463983, 40723336, 90842405, 11291176, 135732256, 90842400, 16], [11370967, 40723320, 90842405, 11291160, 262459440, 90842400, 16], [11370983, 167450520, 90842405, 83589120, 262459456, 90842400, 16], [83463967, 40723336, 90842405, 83589120, 135732256, 90842400, 16], [83463967, 40723336, 5, 11291176, 135732256, 90842400, 16], [167368447, 83589120, 5, 167450520, 135732256, 90842400, 16], [11370983, 40723336, 90842405, 83589136, 135732256, 90842400, 16]], 'aut_group': '16384.ji', 'aut_hash': 3568873400067866534, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 16384, 'aut_permdeg': 24, 'aut_perms': [62949091023446772300149, 458334280099749893096966, 110310246598309326786843, 154906402708027336009871, 131012629009531987024878, 504562820802892200056652, 457182409113072956996017, 518468301460072593053114, 130438581911443332584214, 857377597155548417294, 145831799471596037007949, 146802284752130253271549, 518552471713615257281544, 457815939614251582296726], 'aut_phi_ratio': 256.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 2, 4, 1], [2, 2, 8, 1], [2, 4, 4, 2], [4, 2, 4, 1], [4, 4, 2, 1], [4, 4, 4, 1], [4, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^8.D_4^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '1024.djt', 'autcent_hash': 1957374279625110839, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '16.11', 'autcentquo_hash': 11, 'autcentquo_nilpotent': True, 'autcentquo_order': 16, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 14], [2, 4, 8], [4, 2, 4], [4, 4, 14]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '32.51', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 2216, '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, 14], [2, 4, 1, 8], [4, 2, 1, 4], [4, 4, 1, 14]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 624960, 'exponent': 4, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '32.51', 'hash': 2216, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2, 2, 2, 1, 1], 'inner_gens': [[11370967, 167450536, 90842414, 11291176, 262459456, 90842400, 16], [83463967, 40723336, 90842405, 83589136, 135732256, 90842400, 16], [11370983, 40723336, 90842405, 11291176, 135732256, 90842400, 16], [11370967, 167450536, 90842405, 11291176, 135732256, 90842400, 16], [83463967, 40723336, 90842405, 11291176, 135732256, 90842400, 16], [11370967, 40723336, 90842405, 11291176, 135732256, 90842400, 16], [11370967, 40723336, 90842405, 11291176, 135732256, 90842400, 16]], 'inner_hash': 51, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'C_2^5', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8], [4, 4]], 'label': '128.2216', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'D4^2:C2', 'ngens': 7, 'nilpotency_class': 2, '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': 12, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 44, 'number_divisions': 44, 'number_normal_subgroups': 432, 'number_subgroup_autclasses': 145, 'number_subgroup_classes': 950, 'number_subgroups': 1836, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 63], [4, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 11370967, 0, 219593656, 0, 262459454, 0, 0, 0], 'outer_gens': [[90760327, 212340360, 5, 11291176, 135732256, 90842400, 16], [83463967, 258829936, 135732245, 212340360, 262459440, 90842400, 90842416], [11370967, 40723320, 5, 11291176, 135732240, 90842400, 16], [11370967, 40723320, 135732254, 83589120, 135732256, 90842400, 90842416], [11370967, 40723320, 90842405, 11291160, 262459456, 90842400, 16], [40805407, 212340360, 90842405, 143354296, 135732256, 90842400, 16], [11370967, 40723320, 90842405, 11291176, 135732256, 90842400, 16], [11370967, 40723336, 5, 11291176, 135732256, 90842400, 16], [11370967, 40723320, 90842405, 11291160, 135732256, 90842400, 16]], 'outer_group': '512.6480905', 'outer_hash': 411620650426100056, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 12, 'outer_perms': [40723200, 302376366, 136095263, 6, 40723343, 39916920, 766200, 7, 40723320], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3:D_4^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 8], [4, 4]], 'representations': {'PC': {'code': 37438388643045672, 'gens': [1, 2, 3, 4, 6], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 2019, 1242, 80, 1027, 124]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729074820244, 58301505957300808, 24992896918755244, 25038659807093174, 125101749995522276]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [11040833, 10024001, 32746561, 5128257, 31029334, 19896385, 6210625]}, 'Perm': {'d': 12, 'gens': [11370967, 40723336, 90842405, 11291176, 135732256, 90842400, 16]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4^2:C_2', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
