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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '3888.fx', 'ambient_counter': 154, 'ambient_order': 3888, 'ambient_tex': '(C_3\\times C_6^2):S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 648, 'characteristic': False, 'core_order': 1, 'counter': 1218, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '3888.fx.648.o1.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '648.o1.e1', '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': 648, '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': '6.2', 'subgroup_hash': 2, 'subgroup_order': 6, 'subgroup_tex': 'C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3888.fx', 'aut_centralizer_order': 216, 'aut_label': '648.o1', 'aut_quo_index': None, 'aut_stab_index': 36, 'aut_weyl_group': '2.1', 'aut_weyl_index': 7776, 'centralizer': '6.l1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['216.v1.b1', '216.bb1.b1', '216.be1.a1', '216.be1.f1', '216.bh1.c1', '216.bh1.e1', '216.bh2.a1', '216.bj1.a1', '216.bj1.f1', '324.h1.c1', '324.bz1.e1', '324.bz1.k1'], 'contains': ['1296.f1.c1', '1944.b1.a1'], 'core': '3888.a1.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [8723, -1, 8745, -1, 8972, -1, 9410, -1], 'generators': [324, 2], 'label': '3888.fx.648.o1.e1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.l1.a1', 'old_label': '648.o1.e1', 'projective_image': '3888.fx', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '648.o1.e1', '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [5]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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, 2], [6, 1, 2]], 'center_label': '6.2', 'center_order': 6, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 6]], 'label': '6.2', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 6, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2]], 'representations': {'PC': {'code': 21, 'gens': [1], 'pres': [2, -2, -3, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [73]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 56]}, 'Perm': {'d': 5, 'gens': [24, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6', 'transitive_degree': 6, '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [6, 6, 6, 6, 6, 6, 6], 'aut_gens': [[1, 6, 18, 108, 648], [1525, 1740, 2904, 108, 648], [1745, 1740, 450, 2700, 3240], [2813, 6, 528, 2700, 3240], [3755, 1518, 3120, 2484, 3240], [295, 1308, 3330, 3780, 648], [3545, 1308, 234, 540, 3240], [305, 1518, 318, 540, 3240]], 'aut_group': None, 'aut_hash': 6217189639571012136, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 15552, 'aut_permdeg': 44, 'aut_perms': [2177824797096101636311960760480595465994212293450996244, 2056980415762204710312689524944927561355708307062208955, 968108824159390210110208600316090293925358911052201957, 1570409534852037680073012165177009708189923216229865273, 1860414354071862863603015973879743093779867038190699656, 1984922887271797934588482132545947840030831280165274045, 1568172938434547899129153405904993086941137121664798296], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 1], [2, 9, 2, 1], [2, 18, 1, 1], [2, 54, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 3, 6, 1], [3, 6, 1, 1], [3, 6, 6, 1], [3, 12, 1, 1], [3, 18, 1, 1], [3, 36, 1, 1], [4, 6, 1, 1], [4, 54, 1, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 3], [6, 2, 4, 1], [6, 3, 6, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 6, 6, 2], [6, 6, 12, 1], [6, 9, 4, 1], [6, 9, 12, 1], [6, 12, 1, 2], [6, 12, 2, 1], [6, 18, 1, 1], [6, 18, 2, 3], [6, 18, 4, 2], [6, 18, 6, 2], [6, 18, 12, 2], [6, 36, 1, 3], [6, 36, 2, 1], [6, 54, 2, 1], [6, 54, 6, 1], [6, 108, 1, 1], [9, 18, 2, 1], [9, 36, 2, 1], [12, 6, 2, 1], [12, 18, 6, 1], [12, 36, 1, 1], [12, 54, 2, 1], [12, 54, 6, 1], [12, 108, 1, 1], [18, 18, 2, 1], [18, 36, 2, 2], [18, 36, 4, 1], [18, 108, 2, 1], [36, 108, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3.C_6^2.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '24.15', 'autcent_hash': 15, 'autcent_nilpotent': True, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '648.555', 'autcentquo_hash': 555, 'autcentquo_nilpotent': False, 'autcentquo_order': 648, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3.S_3^3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 6, 1], [2, 9, 2], [2, 18, 1], [2, 54, 1], [3, 1, 2], [3, 2, 3], [3, 3, 6], [3, 6, 7], [3, 12, 1], [3, 18, 1], [3, 36, 1], [4, 6, 1], [4, 54, 1], [6, 1, 2], [6, 2, 11], [6, 3, 6], [6, 6, 27], [6, 9, 16], [6, 12, 4], [6, 18, 51], [6, 36, 5], [6, 54, 8], [6, 108, 1], [9, 18, 2], [9, 36, 2], [12, 6, 2], [12, 18, 6], [12, 36, 1], [12, 54, 8], [12, 108, 1], [18, 18, 2], [18, 36, 8], [18, 108, 2], [36, 108, 2]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '648.602', 'commutator_count': 1, 'commutator_label': '162.48', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 154, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['162.10', 1], ['24.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 1], [2, 9, 1, 2], [2, 18, 1, 1], [2, 54, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 3, 2, 3], [3, 6, 1, 1], [3, 6, 2, 3], [3, 12, 1, 1], [3, 18, 1, 1], [3, 36, 1, 1], [4, 6, 1, 1], [4, 54, 1, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 5], [6, 3, 2, 3], [6, 6, 1, 1], [6, 6, 2, 13], [6, 9, 2, 8], [6, 12, 1, 2], [6, 12, 2, 1], [6, 18, 1, 3], [6, 18, 2, 24], [6, 36, 1, 3], [6, 36, 2, 1], [6, 54, 2, 4], [6, 108, 1, 1], [9, 18, 2, 1], [9, 36, 2, 1], [12, 6, 2, 1], [12, 18, 2, 3], [12, 36, 1, 1], [12, 54, 2, 4], [12, 108, 1, 1], [18, 18, 2, 1], [18, 36, 2, 4], [18, 108, 2, 1], [36, 108, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 943488, 'exponent': 36, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 24]], 'familial': False, 'frattini_label': '18.5', 'frattini_quotient': '216.170', 'hash': 6136387377220627443, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [6, 3, 6, 6, 1], 'inner_gens': [[1, 1740, 1398, 108, 648], [2821, 6, 450, 108, 648], [259, 222, 18, 3780, 648], [1, 6, 882, 108, 648], [1, 6, 18, 108, 648]], 'inner_hash': 602, 'inner_nilpotent': False, 'inner_order': 648, 'inner_split': True, 'inner_tex': 'S_3\\times C_3^2:D_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 6, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 24], [2, 42], [3, 48], [4, 15], [6, 64], [12, 5]], 'label': '3888.fx', 'linC_count': 576, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 336, 'linQ_dim': 10, 'linQ_dim_count': 336, 'linR_count': 336, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': '(C3*C6^2):S3^2', '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 66, 'number_characteristic_subgroups': 82, 'number_conjugacy_classes': 198, 'number_divisions': 113, 'number_normal_subgroups': 90, 'number_subgroup_autclasses': 800, 'number_subgroup_classes': 1307, 'number_subgroups': 13448, 'old_label': None, 'order': 3888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 99], [3, 134], [4, 60], [6, 2034], [9, 108], [12, 696], [18, 540], [36, 216]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1729, 2814, 18, 1836, 648], [3673, 12, 90, 1836, 648], [1589, 2598, 1608, 2700, 3240]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [24, 40320, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [4, 19], [6, 28], [8, 6], [12, 33], [24, 1]], 'representations': {'PC': {'code': '860264766717902947345670872042786850986715954790349779325653', 'gens': [1, 3, 4, 6, 8], 'pres': [9, 2, 3, 3, 2, 3, 2, 3, 2, 3, 18, 46982, 50331, 27768, 2721, 102, 139594, 10678, 10012, 11372, 158, 10617, 214]}, 'Perm': {'d': 16, 'gens': [2895691721883, 280227227283, 1495929981025]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_3\\times C_6^2):S_3^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
