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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '9000.w', 'ambient_counter': 23, 'ambient_order': 9000, 'ambient_tex': '(C_5^2\\times C_{15}):S_4', 'central': False, 'central_factor': False, 'centralizer_order': 25, 'characteristic': False, 'core_order': 375, 'counter': 17, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '9000.w.12.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.b1.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': '750.36', 'subgroup_hash': 36, 'subgroup_order': 750, 'subgroup_tex': 'C_5^2\\times D_{15}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '9000.w', 'aut_centralizer_order': None, 'aut_label': '12.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '360.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '6.c1.a1'], 'contains': ['24.a1.a1', '36.b1.a1', '60.e1.a1', '60.f1.a1', '60.g1.a1', '60.h1.a1', '60.i1.a1'], 'core': '24.a1.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [4301, -1, 3672, -1, 5991, -1, 4646, -1], 'generators': [111253006773504, 52843663923985, 4, 4652849723575704, 16490193360624], 'label': '9000.w.12.b1.a1', 'mobius_quo': None, 'mobius_sub': 2, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.c1.a1', 'old_label': '12.b1.a1', 'projective_image': '9000.w', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.b1.a1', 'subgroup_fusion': None, 'weyl_group': '60.8'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '50.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 120, 'aut_gen_orders': [12, 20, 6, 8], 'aut_gens': [[1, 5, 50], [31, 379, 650], [10, 469, 550], [34, 679, 700], [31, 496, 100]], 'aut_group': None, 'aut_hash': 671234851772411131, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 57600, 'aut_permdeg': 39, 'aut_perms': [11147166436953892388113226913609764266526533027, 17421021209435874167373047791557102506746077533, 12840655702859065455495104043222683134677783163, 11326447158679279458879278363050799867208123795], 'aut_phi_ratio': 288.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 2, 1, 1], [5, 1, 24, 1], [5, 2, 2, 1], [5, 2, 48, 1], [10, 15, 24, 1], [15, 2, 4, 1], [15, 2, 24, 1], [15, 2, 96, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times \\GL(2,5)\\times F_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 120, 'autcent_group': '480.218', 'autcent_hash': 218, 'autcent_nilpotent': False, 'autcent_order': 480, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,5)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '120.36', 'autcentquo_hash': 36, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_5', 'cc_stats': [[1, 1, 1], [2, 15, 1], [3, 2, 1], [5, 1, 24], [5, 2, 50], [10, 15, 24], [15, 2, 124]], 'center_label': '25.2', 'center_order': 25, 'central_product': True, 'central_quotient': '30.3', 'commutator_count': 1, 'commutator_label': '15.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 36, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['30.3', 1], ['5.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 2, 1, 1], [5, 1, 4, 6], [5, 2, 2, 1], [5, 2, 4, 12], [10, 15, 4, 6], [15, 2, 4, 7], [15, 2, 8, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 30, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '750.36', 'hash': 36, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [1, 2, 15], 'inner_gens': [[1, 5, 50], [1, 5, 700], [1, 105, 50]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 30, 'inner_split': True, 'inner_tex': 'D_{15}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 50], [2, 175]], 'label': '750.36', 'linC_count': 3840, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 180, 'linQ_dim': 14, 'linQ_dim_count': 180, 'linR_count': 1920, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5^2*D15', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 225, 'number_divisions': 47, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 112, 'number_subgroups': 320, 'old_label': None, 'order': 750, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 15], [3, 2], [5, 124], [10, 360], [15, 248]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [4, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[23, 35, 650], [44, 29, 50]], 'outer_group': '1920.240616', 'outer_hash': 8108536333046314977, 'outer_nilpotent': False, 'outer_order': 1920, 'outer_permdeg': 28, 'outer_perms': [10999388007287655227557781177, 79248404456545701501070144560], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times \\GL(2,5)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 5, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 13], [8, 19], [16, 12]], 'representations': {'PC': {'code': 21318611492154167, 'gens': [1, 2, 4], 'pres': [5, -5, -2, -5, -3, -5, 26, 2808, 78, 3009]}, 'GLZN': {'d': 2, 'p': 50, 'gens': [125031, 3875031, 125501, 188776, 126599]}, 'Perm': {'d': 18, 'gens': [21010008096000, 454353, 851793, 518918400, 397620186163200]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [5, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5^2\\times D_{15}', 'transitive_degree': 150, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 60, 'aut_gen_orders': [12, 6, 12, 30, 10, 5], 'aut_gens': [[425114656815145, 803536963246947, 47619200730505], [4128363448894205, 2202954865388643, 4552602295662725], [1641884882482109, 5243012248442523, 2516009292489626], [3569200216264826, 4165682140652524, 4864658443592525], [3461596571711786, 782084368558227, 4552602295662722], [3209121772433545, 5096465073725883, 2061038196525625], [2245491139305625, 5262923743869627, 1491275939501065]], 'aut_group': '216000.d', 'aut_hash': 4125830444204071041, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 216000, 'aut_permdeg': 90, 'aut_perms': [1353389823616814961747633978527900157410304644650651683226682096835798062473757528197889397843392306168845821240824209676547277205807687360, 170152806349165417973368187088405632743417951109982953258447029816514047471757083471627302354032916014432967151824536718777043179759725677, 766863171969383379458659654149979975679451565566495273103191375324656774402777058637824507835744502536696969469722902966358404397034235895, 766096427508112839545789622843103692640298224328259753028956003517033109549511623966696498419253736821650874977159182424802010911824116282, 1243263330560354306936515227895462692778893154581448449015769142896529846559830325474506034813954974233547978704974423646357218536864944180, 720717920711527888352902613843913768928128500950180796073031941062491914099714954436871447505207589404317661410551335462125349913488849827], 'aut_phi_ratio': 90.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 75, 1, 1], [2, 90, 1, 1], [3, 2, 1, 1], [3, 200, 3, 1], [4, 2250, 1, 1], [5, 4, 4, 1], [5, 6, 2, 1], [5, 12, 2, 1], [5, 12, 4, 1], [5, 24, 1, 1], [6, 150, 1, 1], [10, 90, 4, 1], [10, 150, 2, 1], [10, 180, 2, 1], [10, 180, 4, 2], [15, 8, 4, 1], [15, 12, 2, 1], [15, 24, 2, 2], [15, 24, 4, 1], [15, 200, 12, 1], [30, 300, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_5^3:\\He_3.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '216000.d', 'autcentquo_hash': 4125830444204071041, 'autcentquo_nilpotent': False, 'autcentquo_order': 216000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_5^3:\\He_3.C_2^3', 'cc_stats': [[1, 1, 1], [2, 75, 1], [2, 90, 1], [3, 2, 1], [3, 200, 3], [4, 2250, 1], [5, 4, 4], [5, 6, 2], [5, 12, 6], [5, 24, 1], [6, 150, 1], [10, 90, 4], [10, 150, 2], [10, 180, 10], [15, 8, 4], [15, 12, 2], [15, 24, 8], [15, 200, 12], [30, 300, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '9000.w', 'commutator_count': 1, 'commutator_label': '4500.r', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 23, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 75, 1, 1], [2, 90, 1, 1], [3, 2, 1, 1], [3, 200, 1, 3], [4, 2250, 1, 1], [5, 4, 4, 1], [5, 6, 2, 1], [5, 12, 2, 1], [5, 12, 4, 1], [5, 24, 1, 1], [6, 150, 1, 1], [10, 90, 4, 1], [10, 150, 2, 1], [10, 180, 2, 1], [10, 180, 4, 2], [15, 8, 4, 1], [15, 12, 2, 1], [15, 24, 2, 2], [15, 24, 4, 1], [15, 200, 4, 3], [30, 300, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1640520, 'exponent': 60, 'exponents_of_order': [3, 3, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 0, 12], [12, -1, 2], [12, 1, 2], [24, 0, 6], [24, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '9000.w', 'hash': 6614943827528876077, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [10, 3, 4], 'inner_gens': [[425114656815145, 487273080934084, 2525390582779705], [884804770125605, 803536963246947, 2932250531834885], [1234716879520921, 2343660514923604, 47619200730505]], 'inner_hash': 6614943827528876077, 'inner_nilpotent': False, 'inner_order': 9000, 'inner_split': True, 'inner_tex': '(C_5^2\\times C_{15}):S_4', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 2], [2, 4], [3, 2], [4, 8], [6, 9], [8, 16], [12, 16], [24, 9]], 'label': '9000.w', 'linC_count': 24, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 6, 'linQ_dim': 14, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': '(C5^2*C15):S4', '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, 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': 24, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 66, 'number_divisions': 28, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 132, 'number_subgroup_classes': 156, 'number_subgroups': 12388, 'old_label': None, 'order': 9000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 165], [3, 602], [4, 2250], [5, 124], [6, 150], [10, 2460], [15, 2648], [30, 600]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[425114656815145, 803536963246948, 47619200730505], [2621896347976082, 3425179599964947, 1696668230690161]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [120, 1450], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 4], [3, 2], [6, 1], [12, 2], [16, 2], [24, 6], [32, 4], [48, 4], [96, 1]], 'representations': {'PC': {'code': '739579673105200760651985666604472286165928704041557357262064396751612602324318454366290797903155327', 'gens': [1, 2, 3, 4, 7, 8], 'pres': [8, -2, -3, -2, 2, -3, -5, 5, 5, 24768, 97985, 55881, 44210, 2242, 96387, 134507, 1235, 91, 10564, 112812, 660, 156, 172813, 2325, 201606, 6062, 13462, 6750, 69127, 34575, 38431]}, 'Perm': {'d': 18, 'gens': [425114656815145, 803536963246947, 47619200730505]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_5^2\\times C_{15}):S_4', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
