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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '3168.c', 'ambient_counter': 3, 'ambient_order': 3168, 'ambient_tex': 'D_{44}:C_{36}', 'central': False, 'central_factor': False, 'centralizer_order': 1584, 'characteristic': False, 'core_order': 3, 'counter': 108, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '3168.c.264.f1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '264.f1.b1', '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': 264, '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': '12.2', 'subgroup_hash': 2, 'subgroup_order': 12, 'subgroup_tex': 'C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3168.c', 'aut_centralizer_order': 1320, 'aut_label': '264.f1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '4.2', 'aut_weyl_index': 5280, 'centralizer': '2.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['24.f1.b1', '88.f1.b1', '132.c1.a1'], 'contains': ['528.b1.a1', '792.f1.b1'], 'core': '1056.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [8280, -1, 1350, -1, 7513, -1, 8433, -1], 'generators': [2127406316, 3941958700, 24778026109], 'label': '3168.c.264.f1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '66.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.a1', 'old_label': '264.f1.b1', 'projective_image': '1056.81', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '264.f1.b1', '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': '12.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2], 'aut_gens': [[1], [5], [7]], 'aut_group': '4.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [1, 6], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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], [4, 1, 2], [6, 1, 2], [12, 1, 4]], 'center_label': '12.2', 'center_order': 12, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 12, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '2.1', '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 12]], 'label': '12.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C12', '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 12, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [12, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5], [7]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [4, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 3865, 'gens': [1], 'pres': [3, -2, -2, -3, 6, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20970031]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [619]}, 'Perm': {'d': 7, 'gens': [2400, 4, 744]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [12], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}', 'transitive_degree': 12, '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': '72.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [2, 10, 30, 20, 30, 10, 10], 'aut_gens': [[13702999292, 15767835078, 158006], [9886182226, 17644958238, 43033609], [8634766984, 500566333, 28893263], [9010191600, 125141745, 40463034], [20272930639, 24340030945, 1320025], [3816817200, 11450451587, 61979640], [5068232659, 8009059127, 57551899], [9635899419, 20961208987, 31363397]], 'aut_group': None, 'aut_hash': 663454934575270710, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21120, 'aut_permdeg': 58, 'aut_perms': [1732732255107955792136087714523083155611527696282609392845413795515060656585530, 1715500695344529927630688436017572204067639240528097053880490743038690386126869, 410949957957637167701383941578475739672622923628152146625556499012530057811211, 23440870890990985064437392413157911354402545286639407489330654141706757358531, 1439559129500943182807148742255642695193111227337758557736047596152297216861084, 1752221241673055078784697114190557739888993379513555775212589631060461638953067, 1766464214964711651262414835527118663334041021630008231056518429352221596925084], 'aut_phi_ratio': 22.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 44, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 4, 1], [4, 44, 1, 1], [6, 1, 2, 1], [6, 2, 2, 1], [6, 44, 2, 1], [8, 44, 2, 1], [9, 1, 6, 1], [11, 2, 5, 1], [12, 1, 4, 1], [12, 2, 2, 1], [12, 2, 8, 1], [12, 44, 2, 1], [18, 1, 6, 1], [18, 2, 6, 1], [18, 44, 6, 1], [22, 2, 5, 1], [22, 2, 10, 1], [24, 44, 4, 1], [33, 2, 10, 1], [36, 1, 12, 1], [36, 2, 6, 1], [36, 2, 24, 1], [36, 44, 6, 1], [44, 2, 10, 2], [44, 2, 40, 1], [66, 2, 10, 1], [66, 2, 20, 1], [72, 44, 12, 1], [99, 2, 30, 1], [132, 2, 20, 2], [132, 2, 80, 1], [198, 2, 30, 1], [198, 2, 60, 1], [396, 2, 60, 2], [396, 2, 240, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{22}.C_{30}.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '48.52', 'autcent_hash': 52, 'autcent_nilpotent': True, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '440.42', 'autcentquo_hash': 42, 'autcentquo_nilpotent': False, 'autcentquo_order': 440, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 44, 1], [3, 1, 2], [4, 1, 2], [4, 2, 5], [4, 44, 1], [6, 1, 2], [6, 2, 2], [6, 44, 2], [8, 44, 2], [9, 1, 6], [11, 2, 5], [12, 1, 4], [12, 2, 10], [12, 44, 2], [18, 1, 6], [18, 2, 6], [18, 44, 6], [22, 2, 15], [24, 44, 4], [33, 2, 10], [36, 1, 12], [36, 2, 30], [36, 44, 6], [44, 2, 60], [66, 2, 30], [72, 44, 12], [99, 2, 30], [132, 2, 120], [198, 2, 90], [396, 2, 360]], 'center_label': '36.2', 'center_order': 36, 'central_product': True, 'central_quotient': '88.5', 'commutator_count': 1, 'commutator_label': '44.2', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '11.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['352.11', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 44, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 2, 2], [4, 44, 1, 1], [6, 1, 2, 1], [6, 2, 2, 1], [6, 44, 2, 1], [8, 44, 2, 1], [9, 1, 6, 1], [11, 2, 5, 1], [12, 1, 4, 1], [12, 2, 2, 1], [12, 2, 4, 2], [12, 44, 2, 1], [18, 1, 6, 1], [18, 2, 6, 1], [18, 44, 6, 1], [22, 2, 5, 1], [22, 2, 10, 1], [24, 44, 4, 1], [33, 2, 10, 1], [36, 1, 12, 1], [36, 2, 6, 1], [36, 2, 12, 2], [36, 44, 6, 1], [44, 2, 10, 2], [44, 2, 20, 2], [66, 2, 10, 1], [66, 2, 20, 1], [72, 44, 12, 1], [99, 2, 30, 1], [132, 2, 20, 2], [132, 2, 40, 2], [198, 2, 30, 1], [198, 2, 60, 1], [396, 2, 60, 2], [396, 2, 120, 2]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 144, 'exponent': 792, 'exponents_of_order': [5, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[2, 0, 240]], 'familial': False, 'frattini_label': '24.9', 'frattini_quotient': '132.7', 'hash': 8813592882577988912, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 44, 'inner_gen_orders': [44, 22, 2], 'inner_gens': [[13702999292, 15767835078, 39829422], [13702999292, 15767835078, 19694773], [312854084, 17644958238, 158006]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 88, 'inner_split': True, 'inner_tex': 'D_{44}', 'inner_used': [1, 3], 'irrC_degree': 2, 'irrQ_degree': 240, 'irrQ_dim': 240, 'irrR_degree': None, 'irrep_stats': [[1, 72], [2, 774]], 'label': '3168.c', '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': 'D44:C36', 'ngens': 3, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 45, 'number_characteristic_subgroups': 57, 'number_conjugacy_classes': 846, 'number_divisions': 51, 'number_normal_subgroups': 57, 'number_subgroup_autclasses': 126, 'number_subgroup_classes': 132, 'number_subgroups': 774, 'old_label': None, 'order': 3168, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 47], [3, 2], [4, 56], [6, 94], [8, 88], [9, 6], [11, 10], [12, 112], [18, 282], [22, 30], [24, 176], [33, 20], [36, 336], [44, 120], [66, 60], [72, 528], [99, 60], [132, 240], [198, 180], [396, 720]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 2, 2, 30], 'outer_gen_pows': [62570774, 62570774, 62570774, 62570774], 'outer_gens': [[19709793593, 7195639040, 158006], [5130803685, 7195639040, 158006], [19084086004, 15767835078, 158006], [10574460798, 500566333, 158006]], 'outer_group': '240.208', 'outer_hash': 208, 'outer_nilpotent': True, 'outer_order': 240, 'outer_permdeg': 16, 'outer_perms': [40279680, 1307674368000, 362880, 6227026593], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{30}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 6], [6, 4], [8, 2], [10, 2], [12, 4], [20, 5], [24, 2], [40, 5], [60, 2], [80, 2], [120, 3], [240, 2]], 'representations': {'PC': {'code': '3976166881536833915045964957510993751887459000011900', 'gens': [1, 2, 7], 'pres': [8, -2, -2, -2, -3, -3, -11, -2, -2, 47841, 41, 48482, 66, 2563, 123, 9604, 156, 34565, 166]}, 'GLFp': {'d': 2, 'p': 397, 'gens': [13702999292, 15767835078, 158006]}, 'Perm': {'d': 28, 'gens': [837608818963245301260293356, 12129846060497475056316969246, 22632721498791788066242577348]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 36], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{44}:C_{36}', 'transitive_degree': 792, 'wreath_data': None, 'wreath_product': False}
