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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1728.34723', 'ambient_counter': 34723, 'ambient_order': 1728, 'ambient_tex': '(C_2\\times C_4).S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 48, 'counter': 130, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.34723.9.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '9.a1.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': 9, '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': '192.1267', 'subgroup_hash': 1267, 'subgroup_order': 192, 'subgroup_tex': 'C_4^2.D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.34723', 'aut_centralizer_order': None, 'aut_label': '9.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '432.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1.a1', '3.b1.a1'], 'contains': ['18.a1.a1', '18.b1.a1', '18.c1.a1', '18.d1.a1', '18.e1.a1', '18.f1.a1', '18.g1.a1', '18.ba1.a1', '18.bb1.a1', '18.bc1.a1', '18.bd1.a1', '18.bo1.a1', '18.bp1.a1', '18.bq1.a1', '18.br1.a1', '27.a1.a1'], 'core': '36.b1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': None, 'generators': [1, 864, 72, 6, 48, 36, 432], 'label': '1728.34723.9.a1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '9.a1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '9.a1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', '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, 16], [9, 2, 4, 80], [97, 2, 4, 80], [1, 10, 4, 80], [1, 98, 4, 80], [1, 2, 12, 80], [1, 2, 100, 80], [1, 2, 4, 88], [1, 2, 4, 176], [1, 2, 4, 80], [65, 2, 4, 16]], 'aut_group': '1536.408526597', 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 1536, '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, 6, 2, 1], [3, 2, 1, 1], [4, 2, 2, 2], [4, 4, 1, 4], [4, 6, 2, 3], [4, 12, 1, 4], [6, 2, 1, 3], [6, 8, 1, 1], [12, 4, 2, 3], [12, 8, 1, 3]], 'aut_supersolvable': None, 'aut_tex': 'C_2^8\\times S_3', '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, 1], [2, 6, 2], [3, 2, 1], [4, 2, 4], [4, 4, 4], [4, 6, 6], [4, 12, 4], [6, 2, 3], [6, 8, 1], [12, 4, 6], [12, 8, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1267, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 6, 1, 2], [3, 2, 1, 1], [4, 2, 2, 2], [4, 4, 1, 4], [4, 6, 1, 2], [4, 6, 2, 2], [4, 12, 1, 4], [6, 2, 1, 3], [6, 8, 1, 1], [12, 4, 2, 3], [12, 8, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 262080, 'exponent': 12, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '48.51', 'hash': 1267, '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, 16], [2, 16], [4, 7]], 'label': '192.1267', 'linC_count': 160, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 64, 'linQ_dim': 10, 'linQ_dim_count': 32, 'linR_count': 32, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.D6', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 91, 'number_conjugacy_classes': 39, 'number_divisions': 32, 'number_normal_subgroups': 95, 'number_subgroup_autclasses': 196, 'number_subgroup_classes': 214, 'number_subgroups': 480, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 19], [3, 2], [4, 108], [6, 14], [12, 48]], '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': '32.51', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 32, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2^5', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 5], [8, 3]], 'representations': {'PC': {'code': 508294637471374793177700440833, 'gens': [1, 2, 3, 5], 'pres': [7, -2, -2, -2, -2, 2, -2, -3, 680, 1059, 58, 3084, 2111, 102, 6725, 124, 6278]}, 'Perm': {'d': 19, 'gens': [6469326637136641, 14230200138629784, 21052349774671200, 26782491680880744, 3767040, 32749658889834240, 3]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_4^2.D_6', 'transitive_degree': 96, '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': '16.14', '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, 12, 144], [73, 10, 60, 720], [865, 10, 12, 720], [1, 82, 60, 720], [1, 866, 12, 144], [1, 2, 84, 144], [1, 10, 876, 720], [1, 10, 12, 792], [1, 10, 60, 1584], [1, 10, 12, 144], [1, 2, 60, 144], [1, 2, 12, 720], [77, 2, 12, 144], [1, 50, 12, 144], [1, 2, 588, 144]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 55296, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 36, 1, 1], [2, 54, 2, 1], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 4, 1, 1], [4, 6, 2, 2], [4, 12, 1, 4], [4, 18, 2, 2], [4, 36, 1, 3], [4, 54, 2, 1], [6, 2, 1, 9], [6, 4, 1, 9], [6, 8, 1, 3], [6, 72, 1, 1], [12, 8, 1, 3], [12, 8, 2, 3], [12, 8, 4, 1], [12, 12, 2, 6], [12, 24, 1, 6], [12, 24, 2, 6], [12, 36, 2, 3], [12, 72, 1, 2]], '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, 36, 1], [2, 54, 2], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 4, 1], [4, 6, 4], [4, 12, 4], [4, 18, 4], [4, 36, 3], [4, 54, 2], [6, 2, 9], [6, 4, 9], [6, 8, 3], [6, 72, 1], [12, 8, 13], [12, 12, 12], [12, 24, 18], [12, 36, 6], [12, 72, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '432.759', 'commutator_count': 1, 'commutator_label': '108.45', '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', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 34723, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 36, 1, 1], [2, 54, 1, 2], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 4, 1, 1], [4, 6, 2, 2], [4, 12, 1, 4], [4, 18, 1, 2], [4, 18, 2, 1], [4, 36, 1, 3], [4, 54, 2, 1], [6, 2, 1, 9], [6, 4, 1, 9], [6, 8, 1, 3], [6, 72, 1, 1], [12, 8, 1, 5], [12, 8, 2, 4], [12, 12, 2, 6], [12, 24, 1, 8], [12, 24, 2, 5], [12, 36, 1, 2], [12, 36, 2, 2], [12, 72, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 44291520, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '432.759', 'hash': 34723, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 432, 'inner_split': None, 'inner_tex': 'C_2\\times S_3^3', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 32], [4, 43], [8, 14]], 'label': '1728.34723', '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': '(C2*C4).S3^3', 'ngens': 9, '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': 78, 'number_characteristic_subgroups': 164, 'number_conjugacy_classes': 105, 'number_divisions': 84, 'number_normal_subgroups': 170, 'number_subgroup_autclasses': 1014, 'number_subgroup_classes': 1156, 'number_subgroups': 9764, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 147], [3, 26], [4, 364], [6, 150], [12, 1040]], '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': '128.2328', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 128, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2^7', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 24], [4, 19], [8, 22], [16, 3]], 'representations': {'PC': {'code': 56989215140319751281060772950816718108388370611719108317878797450825, 'gens': [1, 2, 4, 7], 'pres': [9, 2, 2, 3, 2, 2, 3, 2, 2, 3, 7776, 181, 46, 2162, 506, 16644, 102, 2713, 130, 2606, 68046, 6819, 8349, 186, 8674, 214, 7811]}, 'Perm': {'d': 25, 'gens': [724012180820990298281520, 1244424517780722212390543, 1994133246498593619920056, 2565076951767063286753200, 3155281711748856829440000, 3155281711748829831168000, 325, 22230464256000, 435]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_2\\times C_4).S_3^3', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
