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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1024.dih', 'ambient_counter': 2244, 'ambient_order': 1024, 'ambient_tex': 'C_4^2.C_2\\wr C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 32, 'counter': 40, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1024.dih.16.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '16.g1.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': 16, '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': '64.211', 'subgroup_hash': 211, 'subgroup_order': 64, 'subgroup_tex': 'C_4^2:C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1024.dih', 'aut_centralizer_order': None, 'aut_label': '16.g1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '128.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.h1.a1', '8.i1.a1'], 'contains': ['32.a1.a1', '32.g1.a1', '32.h1.a1', '32.h1.a2', '32.s1.a1', '32.s1.a2', '32.t1.a1', '32.u1.a1', '32.u1.b1'], 'core': '32.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [1652, -1, 3329, -1, 1647, -1, 3324, -1], 'generators': [62, 96, 800, 80], 'label': '1024.dih.16.g1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.b1.a1', 'old_label': '16.g1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.g1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [4, 8, 6], 'aut_gens': [[36873, 40071, 39937, 5193], [39049, 39943, 4161, 37953], [37001, 7311, 7177, 6337], [38921, 7375, 39113, 5129]], 'aut_group': '49152.bbe', 'aut_hash': 2952015372339381660, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 49152, 'aut_permdeg': 32, 'aut_perms': [206412553074368493903095736586652692, 53044018887663722851940784792635402, 11432812583236748412291168696447284], 'aut_phi_ratio': 1536.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 1, 4, 1], [2, 4, 8, 1], [4, 2, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.C_2^7:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 8092891664598236742, 'autcent_nilpotent': True, 'autcent_order': 2048, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8.C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 4, 8], [4, 2, 12]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 211, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['32.34', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 4, 1, 8], [4, 2, 1, 12]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 105, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '16.14', 'hash': 211, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2, 2], 'inner_gens': [[36873, 40071, 39937, 5193], [36873, 40071, 37889, 7369], [36873, 38023, 39937, 5193], [36873, 37895, 39937, 5193]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12]], 'label': '64.211', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, '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': 'C4^2:C2^2', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 28, 'number_divisions': 28, 'number_normal_subgroups': 105, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 249, 'number_subgroups': 441, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 39], [4, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1], 'outer_gen_pows': [39937, 4097, 4097, 4097, 4097, 4097, 4097, 4097, 4097, 4097, 4097, 4097, 4097, 4097, 4097], 'outer_gens': [[36873, 6279, 39937, 5193], [36873, 40071, 36937, 5193], [36873, 6279, 39937, 36937], [36873, 7311, 39937, 5193], [36873, 40071, 7177, 5193], [36873, 40071, 39937, 37953], [38921, 40071, 39937, 5193], [36873, 40071, 39937, 7241], [39049, 40071, 39937, 5193], [36873, 40071, 39937, 7369], [36873, 40071, 38017, 5193], [36873, 40071, 39937, 5193], [36873, 40071, 39937, 5193], [36873, 40071, 39937, 5193], [36873, 40071, 39937, 5193]], 'outer_group': None, 'outer_hash': 7678673216547819622, 'outer_nilpotent': False, 'outer_order': 6144, 'outer_permdeg': 20, 'outer_perms': [5455, 135582469660121141, 19586627938052730, 13075, 268986027100512142, 25991361611717490, 442724486400, 32395454087673600, 1801115993779200, 32390222834995200, 398728658179353600, 0, 0, 0, 0], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^8.S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12]], 'representations': {'PC': {'code': 292129084436, 'gens': [1, 2, 3, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 116, 50, 730, 88]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [706204574960, 705686951426, 141602720900, 706204575934]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [3251, 8645, 8995, 9889, 13613, 1801]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [39937, 37959, 36873, 4161, 39049, 4225]}, 'Perm': {'d': 10, 'gens': [368045, 730929, 730800, 1174440, 1174336, 1174320]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2:C_2^2', 'transitive_degree': 32, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 16, 'aut_gen_orders': [4, 4, 8, 8, 4, 8, 8], 'aut_gens': [[1, 4, 32, 128], [97, 876, 32, 944], [529, 332, 32, 944], [843, 964, 96, 976], [57, 756, 32, 440], [817, 516, 32, 192], [243, 124, 32, 200], [619, 276, 96, 456]], 'aut_group': None, 'aut_hash': 6035287362225783817, 'aut_nilpotency_class': 6, 'aut_nilpotent': True, 'aut_order': 32768, 'aut_permdeg': 384, 'aut_perms': 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121116202515857131693175270394903060287782326590893427296884840546681502782847368983377637768094054727219491726515702673805784498837396501777188262554444986790555631449357518333798713007489182546289972581580364585743475825813619222495595712493318454287667253198833844530375034317958509908908435206370373258256078415499847206318884780246526904798544360876566717527029131620482740436583019525421386916797435100502808850086696990085101693881497931301906515858124252546846274535451060367964363641602104203482897877334686246562581383486586635876468814826593127296784223513731306053954367316877363549980508930334251293702663360564612238381269994632698656074816324761957768297739449627310921295252255686834588472794674837360993842814552903794442529919042471615114499477268268397042357835615768844163484675079505229156758381951695335316], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 32, 1, 1], [2, 64, 2, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 4, 2, 1], [4, 8, 1, 1], [4, 32, 1, 1], [4, 64, 4, 1], [8, 4, 4, 1], [8, 8, 2, 1], [8, 8, 8, 1], [8, 32, 2, 1], [16, 16, 8, 1], [16, 64, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4^3.C_2^4.C_2^5', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': None, 'autcentquo_hash': 6208140336951712643, 'autcentquo_nilpotent': True, 'autcentquo_order': 8192, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3.C_2^6.C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [2, 32, 1], [2, 64, 2], [4, 2, 2], [4, 4, 3], [4, 8, 1], [4, 32, 1], [4, 64, 4], [8, 4, 4], [8, 8, 10], [8, 32, 2], [16, 16, 8], [16, 64, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '512.1838', 'commutator_count': 2, 'commutator_label': '128.179', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 2244, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 32, 1, 1], [2, 64, 1, 2], [4, 2, 1, 2], [4, 4, 1, 3], [4, 8, 1, 1], [4, 32, 1, 1], [4, 64, 2, 2], [8, 4, 2, 2], [8, 8, 1, 2], [8, 8, 2, 4], [8, 32, 2, 1], [16, 16, 2, 4], [16, 64, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 16, 'exponents_of_order': [10], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[8, 1, 8]], 'familial': False, 'frattini_label': '256.2844', 'frattini_quotient': '4.2', 'hash': 505083576705387077, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [8, 8, 2, 8], 'inner_gens': [[1, 148, 32, 440], [977, 4, 96, 160], [1, 68, 32, 128], [809, 100, 32, 128]], 'inner_hash': 1034948244345798672, 'inner_nilpotent': True, 'inner_order': 512, 'inner_split': None, 'inner_tex': 'C_4^2.C_4\\wr C_2', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 10], [4, 17], [8, 11]], 'label': '1024.dih', 'linC_count': 8, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 4, 'linQ_dim': 16, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.C2wrC4', 'ngens': 2, 'nilpotency_class': 7, 'nilpotent': True, 'normal_counts': [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': 19, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 46, 'number_divisions': 30, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 213, 'number_subgroup_classes': 363, 'number_subgroups': 3813, 'old_label': None, 'order': 1024, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 167], [4, 312], [8, 160], [16, 384]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [96, 312, 362, 112, 608], 'outer_gens': [[625, 36, 32, 192], [643, 28, 32, 136], [73, 660, 32, 504], [353, 532, 32, 640], [859, 468, 96, 400]], 'outer_group': '64.193', 'outer_hash': 193, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 32, 'outer_perms': [61359211852608767692984943193077998, 65480704795541004387875626219965596, 113324326726187840421577805918431882, 225069911148678500642314332909604560, 52387529969665291155189941225975086], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 32, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 9], [8, 9], [16, 4]], 'representations': {'PC': {'code': '237605593520588123342671235491883536806422556954066425666572975069626498067439039', 'gens': [1, 3, 6, 8], 'pres': [10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 20, 971, 4442, 912, 82, 14403, 1773, 113, 29604, 2014, 244, 1465, 175, 35207, 39697, 3227, 237, 70568, 37458, 7228, 268, 57609]}, 'Perm': {'d': 32, 'gens': [76209879730140522703248896331659647, 93185859240955738096818815502524759]}}, 'schur_multiplier': [2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.C_2\\wr C_4', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
