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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '512.519620', 'ambient_counter': 519620, 'ambient_order': 512, 'ambient_tex': 'C_2^6:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 256, 'characteristic': False, 'core_order': 32, 'counter': 105, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '512.519620.16.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '16.f1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '16.14', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 14, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^4', 'simple': False, 'solvable': True, 'special_labels': ['C4'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '32.51', 'subgroup_hash': 51, 'subgroup_order': 32, 'subgroup_tex': 'C_2^5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '512.519620', 'aut_centralizer_order': None, 'aut_label': '16.f1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2.d1', 'complements': [], 'conjugacy_class_count': 3, 'contained_in': ['8.b1', '8.c1', '8.d1', '8.e1', '8.f1', '8.g1', '8.h1', '8.i1'], 'contains': ['32.a1', '32.g1', '32.h1', '32.i1', '32.m1', '32.q1', '32.br1'], 'core': '16.f1', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [11, 4, 352, 256, 320], 'label': '512.519620.16.f1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '16.f1', 'normal_contained_in': ['8.b1', '8.c1', '8.d1', '8.e1', '8.f1', '8.g1', '8.h1', '8.i1'], 'normal_contains': ['32.a1', '32.g1', '32.h1', '32.i1', '32.m1'], 'normalizer': '1.a1', 'old_label': '16.f1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.f1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 26040, 'aut_gen_orders': [2, 5], 'aut_gens': [[1, 2, 4, 8, 16], [5, 2, 4, 12, 16], [11, 8, 27, 26, 13]], 'aut_group': '9999360.a', 'aut_hash': 2505043529366670169, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9999360, 'aut_permdeg': 31, 'aut_perms': [2450617681096434660778009755366, 945758899833610526331495647332850], 'aut_phi_ratio': 624960.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 31, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(5,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 26040, 'autcent_group': '9999360.a', 'autcent_hash': 2505043529366670169, 'autcent_nilpotent': False, 'autcent_order': 9999360, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(5,2)', '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, 31]], 'center_label': '32.51', 'center_order': 32, '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', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 51, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 5]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 31]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3, 5, 7, 31], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '32.51', 'hash': 51, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8, 16], [1, 2, 4, 8, 16], [1, 2, 4, 8, 16], [1, 2, 4, 8, 16], [1, 2, 4, 8, 16]], '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, 32]], 'label': '32.51', 'linC_count': None, 'linC_degree': 5, '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': 'C2^5', 'ngens': 5, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 32, 'number_divisions': 32, 'number_normal_subgroups': 374, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 374, 'number_subgroups': 374, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 31]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 26040, 'outer_gen_orders': [2, 5], 'outer_gen_pows': [0, 0], 'outer_gens': [[5, 2, 4, 12, 16], [11, 8, 27, 26, 13]], 'outer_group': '9999360.a', 'outer_hash': 2505043529366670169, 'outer_nilpotent': False, 'outer_order': 9999360, 'outer_permdeg': 31, 'outer_perms': [2450617681096434660778009755366, 945758899833610526331495647332850], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(5,2)', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4, 5], 'pres': [5, -2, 2, 2, 2, 2]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 705686952884, 706461793860, 706460730980, 706461792404]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [5226561, 7292993, 5128257, 20225089, 6210625]}, 'Perm': {'d': 10, 'gens': [362880, 5040, 120, 6, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^5', '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [12, 2, 4, 4, 6, 6], 'aut_gens': [[1, 2, 8, 16, 64, 128, 256], [13, 174, 268, 59, 36, 43, 320], [357, 326, 332, 276, 64, 388, 256], [105, 110, 76, 283, 64, 448, 292], [321, 173, 12, 148, 36, 367, 260], [1, 226, 332, 308, 36, 303, 356], [37, 169, 76, 123, 68, 495, 256]], 'aut_group': None, 'aut_hash': 2211067084685846733, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 50331648, 'aut_permdeg': 96, 'aut_perms': [106403760560088702279747202928864106569492072709237840565140128491619462324481274593932440136000429644613745070083034043872455635119199832337538299273, 741341701994748282824162646533815743505562499015686137685571086470731127817257244345688621841138918377504880121204153658381702710498763530374586662646, 460427070826361911977204198755493041883265142455046189407773922432143487258928919986167385707631387675858154063900501728937480658876992680259164016128, 682054237289259886870096847170340604548016378772382249020059258959379520493915125138525553636224892223328407590920874191317206562690303734889449689990, 878158957657156659487864721885741571558918891893233131482743151874177139271257916794042229487474916557641188436823530933408019680755670105547546519984, 193933825519758929121684191044213746640605004063733298181613718864477424749270751920642164732713590606207759774596737847360595933005662480599146971304], 'aut_phi_ratio': 196608.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 3, 2], [2, 1, 8, 1], [2, 2, 8, 1], [2, 2, 24, 1], [2, 4, 12, 1], [2, 4, 16, 1], [2, 8, 4, 1], [4, 4, 16, 1], [4, 8, 12, 1], [4, 16, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^{15}.A_4.C_2^6.C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': None, 'autcent_hash': 5043351970803840529, 'autcent_nilpotent': True, 'autcent_order': 1048576, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{20}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '48.48', 'autcentquo_hash': 48, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 15], [2, 2, 32], [2, 4, 28], [2, 8, 4], [4, 4, 16], [4, 8, 12], [4, 16, 8]], 'center_label': '16.14', 'center_order': 16, 'central_product': False, 'central_quotient': '32.51', 'commutator_count': 1, 'commutator_label': '16.14', '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'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 519620, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [2, 2, 1, 32], [2, 4, 1, 28], [2, 8, 1, 4], [4, 4, 1, 16], [4, 8, 1, 12], [4, 16, 1, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 208320, 'exponent': 4, 'exponents_of_order': [9], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.14', 'frattini_quotient': '32.51', 'hash': 519620, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2, 2, 1, 2, 1], 'inner_gens': [[1, 6, 8, 368, 64, 224, 256], [5, 2, 8, 336, 64, 224, 256], [1, 2, 8, 48, 64, 128, 256], [353, 322, 40, 16, 64, 128, 256], [1, 2, 8, 16, 64, 128, 256], [97, 98, 8, 16, 64, 128, 256], [1, 2, 8, 16, 64, 128, 256]], 'inner_hash': 51, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'C_2^5', 'inner_used': [1, 2, 3, 4, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 72], [4, 12]], 'label': '512.519620', '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': 'C2^6:D4', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 13, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 116, 'number_divisions': 116, 'number_normal_subgroups': 1218, 'number_subgroup_autclasses': 485, 'number_subgroup_classes': 34026, 'number_subgroups': 96242, 'old_label': None, 'order': 512, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 223], [4, 288]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [4, 6, 4, 4, 4], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[5, 6, 332, 116, 68, 139, 260], [257, 173, 12, 347, 68, 207, 256], [257, 354, 360, 59, 64, 416, 260], [269, 334, 8, 191, 64, 452, 324], [257, 322, 264, 276, 68, 395, 260]], 'outer_group': None, 'outer_hash': 3654333158879926836, 'outer_nilpotent': False, 'outer_order': 1572864, 'outer_permdeg': 72, 'outer_perms': [983626887595508958437758228756278270146499816500913812432344763496507585892962801037613172204942584316, 11189357695028280834747007866597844355692026221367459422624478015158538247544467169599340684297776485684, 2637339497605485699525565615030109546380015937758592849341020365353730120484794840238726906133529909593, 3498273947015198257944594069899553947633294191969319736262855468370844162155443677929808693029935681632, 922543538657226959379199959882209607313333307265866078189611986500231225041357859570672502882514963736], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^{12}.A_4.C_2.C_2^4', 'pc_rank': None, 'perfect': False, 'permutation_degree': None, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 72], [4, 12]], 'representations': {'PC': {'code': '1301880275822590247390678066940178', 'gens': [1, 2, 4, 5, 7, 8, 9], 'pres': [9, 2, 2, 2, 2, 2, 2, 2, 2, 2, 109, 46, 16564, 7573, 301, 130, 16135, 8080]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_2^6:D_4', 'transitive_degree': 64, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 420, 'aut_gen_orders': [6, 3], 'aut_gens': [[1, 2, 4, 8], [12, 4, 5, 3], [10, 14, 7, 5]], 'aut_group': '20160.a', 'aut_hash': 3764836782182912467, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20160, 'aut_permdeg': 8, 'aut_perms': [5193, 5760], 'aut_phi_ratio': 2520.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 15, 1]], 'aut_supersolvable': False, 'aut_tex': 'A_8', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 420, 'autcent_group': '20160.a', 'autcent_hash': 3764836782182912467, 'autcent_nilpotent': False, 'autcent_order': 20160, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'A_8', '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, 15]], 'center_label': '16.14', 'center_order': 16, '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', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 4]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '16.14', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]], '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, 16]], 'label': '16.14', 'linC_count': 840, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 840, 'linQ_dim': 4, 'linQ_dim_count': 840, 'linR_count': 840, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 16, 'number_divisions': 16, 'number_normal_subgroups': 67, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 67, 'number_subgroups': 67, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 420, 'outer_gen_orders': [6, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[12, 4, 5, 3], [10, 14, 7, 5]], 'outer_group': '20160.a', 'outer_hash': 3764836782182912467, 'outer_nilpotent': False, 'outer_order': 20160, 'outer_permdeg': 8, 'outer_perms': [5193, 5760], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'A_8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4], 'pres': [4, -2, 2, 2, 2]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7115648, 35812976, 7115160]}, 'GLFp': {'d': 4, 'p': 2, 'gens': [33837, 18465, 27183, 18467]}, 'Perm': {'d': 8, 'gens': [5040, 120, 6, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
