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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '192.658', 'ambient_counter': 658, 'ambient_order': 192, 'ambient_tex': 'C_2\\times C_{12}.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 8, 'counter': 58, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '192.658.12.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '12.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': 12, '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': '16.2', 'subgroup_hash': 2, 'subgroup_order': 16, 'subgroup_tex': 'C_4^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '192.658', 'aut_centralizer_order': 64, 'aut_label': '12.e1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '8.5', 'aut_weyl_index': 384, 'centralizer': '6.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.d1.a1', '6.b1.a1', '6.d1.a1', '6.d1.b1'], 'contains': ['24.b1.a1', '24.f1.a1', '24.g1.a1'], 'core': '24.b1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [2, 8], 'label': '192.658.12.e1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '12.e1.a1', 'projective_image': '24.14', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.e1.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': '16.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 6, 2, 2], 'aut_gens': [[1, 4], [3, 5], [3, 12], [14, 13], [9, 6], [3, 14]], 'aut_group': '96.195', 'aut_hash': 195, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [134, 16, 1447, 11520, 5160], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [4, 1, 12, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,\\mathbb{Z}/4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '96.195', 'autcent_hash': 195, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,\\mathbb{Z}/4)', '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], [4, 1, 12]], 'center_label': '16.2', 'center_order': 16, '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', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', '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, 4], [1, 4]], '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, 16]], 'label': '16.2', 'linC_count': 48, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 12, 'linQ_dim': 4, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 16, 'number_divisions': 10, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 15, 'number_subgroups': 15, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[3, 5], [3, 12], [14, 13], [9, 6], [3, 14]], 'outer_group': '96.195', 'outer_hash': 195, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [134, 16, 1447, 11520, 5160], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,\\mathbb{Z}/4)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6]], 'representations': {'PC': {'code': 10245, 'gens': [1, 3], 'pres': [4, 2, 2, 2, 2, 8, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16917782, 35931238]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 377]}, 'Perm': {'d': 8, 'gens': [16560, 22, 5160, 7]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2', 'transitive_degree': 16, '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': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 12, 2, 2, 4, 12, 12], 'aut_gens': [[1, 8, 32], [5, 72, 176], [103, 76, 36], [97, 120, 160], [115, 156, 160], [101, 124, 36], [101, 88, 48], [119, 88, 52]], 'aut_group': None, 'aut_hash': 1317301751931807021, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3072, 'aut_permdeg': 40, 'aut_perms': [269566422812011700910377214569621539715474125506, 698155741755499930889635167380747065326079011767, 455343512984719954345295438112994679587288449811, 329114338466174616987417044037563438307324794747, 686654780890694849146093016608113826050152343252, 693466715578797623984109989156058217525240003875, 596593083755174939271835597757105480238106770773], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 4, 1], [3, 2, 1, 1], [4, 1, 2, 2], [4, 1, 4, 1], [4, 6, 4, 2], [6, 2, 1, 3], [6, 2, 4, 1], [8, 2, 8, 1], [8, 6, 8, 1], [12, 2, 2, 2], [12, 2, 4, 1], [24, 2, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^6.C_2^4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 4817036966848599930, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5.C_2^4', 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': 6, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 2, 1], [4, 1, 8], [4, 6, 8], [6, 2, 7], [8, 2, 8], [8, 6, 8], [12, 2, 8], [24, 2, 16]], 'center_label': '16.10', 'center_order': 16, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', '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': 658, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['96.21', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [4, 1, 2, 4], [4, 6, 1, 4], [4, 6, 2, 2], [6, 2, 1, 7], [8, 2, 4, 2], [8, 6, 4, 2], [12, 2, 2, 4], [24, 2, 8, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 672, 'exponent': 24, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '24.14', 'hash': 658, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 3], 'inner_gens': [[1, 24, 32], [17, 8, 160], [1, 72, 32]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': None, 'inner_tex': 'D_6', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 40]], 'label': '192.658', '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*C12.D4', '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': 21, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 72, 'number_divisions': 36, 'number_normal_subgroups': 87, 'number_subgroup_autclasses': 78, 'number_subgroup_classes': 138, 'number_subgroups': 248, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 56], [6, 14], [8, 64], [12, 16], [24, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4, 2, 2, 4, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[103, 104, 52], [7, 12, 52], [7, 12, 32], [113, 8, 36], [99, 12, 32]], 'outer_group': '256.27633', 'outer_hash': 27633, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 64, 'outer_perms': [84417378081239837509918683087515522346491598570894701836828949747662470117704061080115137, 15966454821041384395072948788529373656845528474017517199024291976907895442607811272514266, 47964364203442446402379804898014211168567384529031947717113849697738404019390824841360703, 88925468610214803895882741592707839317889130982583537424444170212880363424325175191059630, 114886537239023184838277742826558758949073994520196719640913072957011654040631453468015349], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 12], [8, 4]], 'representations': {'PC': {'code': 869756614562718620358017, 'gens': [1, 4, 6], 'pres': [7, -2, -2, -2, -2, -2, -2, -3, 14, 36, 675, 80, 866, 124, 811]}, 'GLZN': {'d': 2, 'p': 20, 'gens': [136017, 8201, 72009, 110467, 24003, 11293, 8189]}, 'Perm': {'d': 17, 'gens': [22317682838407, 44460928512001, 21009969038455, 1271166, 44460928512000, 374406, 518918400]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], '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_{12}.D_4', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
