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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '540.60', 'ambient_counter': 60, 'ambient_order': 540, 'ambient_tex': 'C_6^2.C_{15}', 'central': False, 'central_factor': False, 'centralizer_order': 180, 'characteristic': False, 'core_order': 9, 'counter': 25, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '540.60.30.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '30.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': 30, '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': '18.5', 'subgroup_hash': 5, 'subgroup_order': 18, 'subgroup_tex': 'C_3\\times C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '540.60', 'aut_centralizer_order': 144, 'aut_label': '30.a1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '6.1', 'aut_weyl_index': 432, 'centralizer': '3.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.a1.a1', '15.a1.a1'], 'contains': ['60.a1.a1', '90.a1.a1', '90.b1.a1', '90.b1.b1', '90.b1.c1'], 'core': '60.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [7518, -1, 5537, -1, 4757, -1, 5954, -1], 'generators': [279, 6, 360], 'label': '540.60.30.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '15.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '30.a1.a1', 'projective_image': '180.31', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '30.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 3], [1, 17], [7, 3]], 'aut_group': '48.29', 'aut_hash': 29, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 8, 'aut_perms': [475, 23888], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 8, 1], [6, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', '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, 1], [3, 1, 8], [6, 1, 8]], 'center_label': '18.5', 'center_order': 18, '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', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [6, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '18.5', 'hash': 5, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 18]], 'label': '18.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 18, 'linQ_dim': 4, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C6', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 18, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 12, 'number_subgroups': 12, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 8], [6, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 17], [7, 3]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [475, 23888], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8]], 'representations': {'PC': {'code': 277, 'gens': [1, 2], 'pres': [3, -3, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931151, 16858245]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 1374]}, 'Perm': {'d': 8, 'gens': [5040, 240, 4]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_6', 'transitive_degree': 18, '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': '45.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [3, 3, 4, 6, 3, 3, 2, 2], 'aut_gens': [[1, 3, 18], [361, 3, 18], [1, 183, 18], [1, 3, 126], [2, 3, 207], [13, 3, 18], [1, 282, 27], [10, 3, 18], [271, 3, 18]], 'aut_group': None, 'aut_hash': 5288594478412682658, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2592, 'aut_permdeg': 76, 'aut_perms': [1836891382903975804459394001091160723926710335399602128915795394919470365413243167199681830741257438972554171717, 1786593459177191171708144447917261803556114104410425210469816765935039265604728045855629774591843833494501074744, 2926939416592054301938171105868160, 1340364366935166845357662547534006424267606397746149028107813743310923519565752316665815004688832102892330497274, 233301774675688348215263378000204358007483458123599062349211715282982384264335240103536713501957735765706702490, 1364838065383573004845281734106910110198736141340790277881391852531465080862167172595269806504693490210006887000, 747064279059144484804749504803223517661102711776021428035524451459340625891486219822921361293873310974585697690, 1384729416559188139180349033802711391428160935236675486737159453591660837566346475483270663560412265555365386453], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 3, 1, 2], [5, 1, 4, 1], [6, 3, 2, 1], [6, 3, 3, 2], [9, 12, 6, 1], [10, 3, 4, 1], [15, 1, 8, 1], [15, 3, 4, 2], [30, 3, 8, 1], [30, 3, 12, 2], [45, 12, 24, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_3^2\\times A_4).D_6.C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '36.8', 'autcent_hash': 8, 'autcent_nilpotent': True, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3\\times C_{12}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '72.43', 'autcentquo_hash': 43, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3:S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 1, 2], [3, 3, 2], [5, 1, 4], [6, 3, 8], [9, 12, 6], [10, 3, 4], [15, 1, 8], [15, 3, 8], [30, 3, 32], [45, 12, 24]], 'center_label': '15.1', 'center_order': 15, 'central_product': True, 'central_quotient': '36.11', 'commutator_count': 1, 'commutator_label': '12.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 60, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['108.21', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [5, 1, 4, 1], [6, 3, 2, 4], [9, 12, 2, 3], [10, 3, 4, 1], [15, 1, 8, 1], [15, 3, 8, 1], [30, 3, 8, 4], [45, 12, 8, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 90, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 24]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '180.31', 'hash': 60, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 6, 2], 'inner_gens': [[1, 102, 27], [460, 3, 18], [10, 3, 18]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'C_3\\times A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 6, 'irrep_stats': [[1, 45], [3, 55]], 'label': '540.60', 'linC_count': 24, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 3, 'linQ_dim': 10, 'linQ_dim_count': 3, 'linR_count': 12, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6^2.C15', 'ngens': 6, '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': 18, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 100, 'number_divisions': 22, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 44, 'number_subgroups': 100, 'old_label': None, 'order': 540, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 3], [3, 8], [5, 4], [6, 24], [9, 72], [10, 12], [15, 32], [30, 96], [45, 288]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 12], 'outer_gen_pows': [6, 0, 0], 'outer_gens': [[2, 3, 207], [1, 183, 18], [7, 183, 126]], 'outer_group': '72.32', 'outer_hash': 32, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 10, 'outer_perms': [40440, 404040, 849], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:S_3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [3, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [3, 1], [4, 1], [6, 5], [8, 4], [12, 1], [24, 5]], 'representations': {'PC': {'code': 12781422238649035522015474759, 'gens': [1, 2, 4], 'pres': [6, 3, 2, 3, 2, 3, 5, 2160, 1225, 31, 3350, 651, 69, 118]}, 'Perm': {'d': 18, 'gens': [22230464308797, 16371, 6749568000, 104197, 356995102464000, 753220435968000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 15], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2.C_{15}', 'transitive_degree': 90, 'wreath_data': None, 'wreath_product': False}
