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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '960.10192', 'ambient_counter': 10192, 'ambient_order': 960, 'ambient_tex': 'C_5\\times D_{12}:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': False, 'core_order': 80, 'counter': 65, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.10192.6.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '6.c1.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': 6, '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': '160.182', 'subgroup_hash': 182, 'subgroup_order': 160, 'subgroup_tex': 'C_{20}:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.10192', 'aut_centralizer_order': 8, 'aut_label': '6.c1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '256.56082', 'aut_weyl_index': 48, 'centralizer': '48.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1', '3.a1.a1'], 'contains': ['12.a1.a1', '12.g1.b1', '12.h1.a1', '12.j1.a1', '12.k1.a1', '12.q1.b1', '12.w1.a1', '30.c1.a1'], 'core': '12.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [3219, -1, 3012, -1, 5520, -1, 5142, -1], 'generators': [1, 480, 12, 192, 242, 504], 'label': '960.10192.6.c1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '6.c1.a1', 'projective_image': '48.51', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.c1.a1', 'subgroup_fusion': None, 'weyl_group': '16.14'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 2, 4, 2, 2, 2, 2], 'aut_gens': [[1, 2, 8], [81, 126, 24], [1, 2, 92], [85, 126, 12], [81, 6, 8], [1, 6, 8], [85, 86, 8], [1, 2, 76]], 'aut_group': '512.7532443', 'aut_hash': 3711438873667065943, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 16, 'aut_perms': [8028190211776, 3139424028847, 1596153444480, 5599050065296, 4464821450880, 3052245374767, 2965060188847], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 4, 2, 1], [4, 2, 2, 2], [4, 4, 2, 1], [5, 1, 4, 1], [10, 1, 4, 3], [10, 2, 8, 1], [10, 4, 8, 1], [20, 2, 8, 2], [20, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3.C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.56082', 'autcent_hash': 56082, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\\times C_4', '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, 3], [2, 2, 2], [2, 4, 2], [4, 2, 4], [4, 4, 2], [5, 1, 4], [10, 1, 12], [10, 2, 8], [10, 4, 8], [20, 2, 16], [20, 4, 8]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '8.5', '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', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 182, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['32.28', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 2], [5, 1, 4, 1], [10, 1, 4, 3], [10, 2, 4, 2], [10, 4, 4, 2], [20, 2, 4, 2], [20, 2, 8, 1], [20, 4, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 2604, 'exponent': 20, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '40.14', 'hash': 182, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[1, 6, 8], [5, 2, 88], [1, 82, 8]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 30]], 'label': '160.182', 'linC_count': 288, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 32, 'linQ_dim': 8, 'linQ_dim_count': 32, 'linR_count': 96, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C20:D4', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 18, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 70, 'number_divisions': 26, 'number_normal_subgroups': 48, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 94, 'number_subgroups': 148, 'old_label': None, 'order': 160, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 15], [4, 16], [5, 4], [10, 60], [20, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [8, 0, 0, 0, 0], 'outer_gens': [[1, 42, 8], [1, 2, 12], [1, 2, 88], [81, 2, 8], [1, 2, 136]], 'outer_group': '64.260', 'outer_hash': 260, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 12, 'outer_perms': [39916800, 362880, 5040, 120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 9], [8, 4], [16, 1]], 'representations': {'PC': {'code': 20055205563678826499, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -2, -2, -2, -5, 73, 31, 1065, 69, 88]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [898450, 826574, 359380, 35960, 794497, 28083]}, 'Perm': {'d': 13, 'gens': [83503440, 81360, 482762160, 96, 131760, 1037836800]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}:D_4', 'transitive_degree': 80, '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': '80.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[1, 2, 4, 48], [505, 2, 20, 48], [481, 2, 20, 48], [1, 506, 20, 48], [1, 482, 20, 48], [1, 2, 524, 48], [1, 2, 500, 48], [1, 2, 20, 552], [1, 2, 20, 528], [1, 2, 20, 624], [1, 2, 260, 432], [1, 2, 20, 48], [1, 18, 4, 48]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 12288, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 4, 2, 1], [2, 6, 2, 1], [2, 6, 4, 1], [3, 2, 1, 1], [4, 2, 2, 2], [4, 4, 2, 1], [4, 6, 2, 1], [4, 12, 1, 2], [4, 12, 2, 1], [5, 1, 4, 1], [6, 2, 1, 3], [6, 4, 2, 1], [6, 8, 2, 1], [10, 1, 4, 3], [10, 4, 4, 1], [10, 4, 8, 1], [10, 6, 8, 1], [10, 6, 16, 1], [12, 4, 2, 2], [12, 8, 2, 1], [15, 2, 4, 1], [20, 2, 8, 2], [20, 4, 8, 1], [20, 6, 8, 1], [20, 12, 4, 2], [20, 12, 8, 1], [30, 2, 4, 3], [30, 4, 8, 1], [30, 8, 8, 1], [60, 4, 8, 2], [60, 8, 8, 1]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 3], [2, 6, 6], [3, 2, 1], [4, 2, 4], [4, 4, 2], [4, 6, 2], [4, 12, 4], [5, 1, 4], [6, 2, 3], [6, 4, 2], [6, 8, 2], [10, 1, 12], [10, 4, 12], [10, 6, 24], [12, 4, 4], [12, 8, 2], [15, 2, 4], [20, 2, 16], [20, 4, 8], [20, 6, 8], [20, 12, 16], [30, 2, 12], [30, 4, 8], [30, 8, 8], [60, 4, 16], [60, 8, 8]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '48.51', 'commutator_count': 1, 'commutator_label': '12.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10192, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['192.1171', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 3], [2, 6, 1, 6], [3, 2, 1, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 2], [4, 6, 2, 1], [4, 12, 1, 4], [5, 1, 4, 1], [6, 2, 1, 3], [6, 4, 2, 1], [6, 8, 1, 2], [10, 1, 4, 3], [10, 4, 4, 3], [10, 6, 4, 6], [12, 4, 1, 2], [12, 4, 2, 1], [12, 8, 1, 2], [15, 2, 4, 1], [20, 2, 4, 2], [20, 2, 8, 1], [20, 4, 4, 2], [20, 6, 8, 1], [20, 12, 4, 4], [30, 2, 4, 3], [30, 4, 8, 1], [30, 8, 4, 2], [60, 4, 4, 2], [60, 4, 8, 1], [60, 8, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 20442240, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '240.206', 'hash': 10192, 'hyperelementary': 2, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 48, 'inner_split': None, 'inner_tex': 'C_2^2\\times D_6', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 80], [4, 35]], 'label': '960.10192', 'linC_count': 16640, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 1632, 'linQ_dim': 12, 'linQ_dim_count': 1632, 'linR_count': 272, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5*D12:D4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 48, 'number_characteristic_subgroups': 86, 'number_conjugacy_classes': 195, 'number_divisions': 70, 'number_normal_subgroups': 210, 'number_subgroup_autclasses': 364, 'number_subgroup_classes': 668, 'number_subgroups': 1984, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 51], [3, 2], [4, 76], [5, 4], [6, 30], [10, 204], [12, 32], [15, 8], [20, 304], [30, 120], [60, 128]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '256.55630', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 256, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_4^2:C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 21], [8, 14], [16, 5], [32, 2]], 'representations': {'PC': {'code': 10948691830887240442545770156050404420091971, 'gens': [1, 2, 3, 6], 'pres': [8, -2, -2, -2, -2, -3, -2, -2, -5, 8097, 674, 6298, 66, 651, 91, 652, 6357, 141, 166]}, 'GLZN': {'d': 2, 'p': 110, 'gens': [99403164, 1332299, 22757927, 118459089, 107811081, 145079109, 109162378, 75207551]}, 'Perm': {'d': 20, 'gens': [363600, 123111177771859200, 21027497243114880, 268987316137344000, 33, 385921040067763200, 385921040108042880, 5760]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_{12}:D_4', 'transitive_degree': 240, 'wreath_data': None, 'wreath_product': False}
