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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '640.19151', 'ambient_counter': 19151, 'ambient_order': 640, 'ambient_tex': 'D_{10}.(C_4\\times D_4)', 'central': False, 'central_factor': False, 'centralizer_order': 160, 'characteristic': True, 'core_order': 16, 'counter': 210, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '640.19151.40.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '40.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '40.12', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 12, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 40, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times F_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '16.14', 'subgroup_hash': 14, 'subgroup_order': 16, 'subgroup_tex': 'C_2^4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '640.19151', 'aut_centralizer_order': None, 'aut_label': '40.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '4.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.a1', '20.a1', '20.b1', '20.c1'], 'contains': ['80.a1', '80.b1', '80.j1', '80.z1', '80.bp1'], 'core': '40.a1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [64801, 65031, 1984831, 1344021], 'label': '640.19151.40.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '40.a1', 'normal_contained_in': ['8.a1', '20.a1'], 'normal_contains': ['80.a1', '80.b1'], 'normalizer': '1.a1', 'old_label': '40.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '40.a1', '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': '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}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 2, 4, 4, 10, 2, 4], 'aut_gens': [[64801, 576071, 1984831, 1504171, 112891], [1344821, 1856691, 1984831, 2145141, 720241], [64801, 1216941, 1984831, 864481, 112731], [64801, 1217581, 1984831, 1504651, 720561], [1984031, 576071, 1984831, 1504971, 113051], [1344821, 2496761, 1984831, 225471, 1392911], [64801, 1857011, 1984831, 224031, 112731], [1344821, 576871, 1984831, 1504971, 721361]], 'aut_group': None, 'aut_hash': 8212277918716637030, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 40960, 'aut_permdeg': 56, 'aut_perms': [661514423886058076216013207114250184720797799961372902167526092463753162026, 672325676679091313302178013864311740851464373907881859486103949727039473048, 710776475723053259034951110771202803528151569378626426914663383424479949943, 84965906191809193816465867327727704641010307307201044137330869906572445627, 493187535359414995274039201424211024265213134496050459098484834230733481162, 168032451326454673599013597133284111852313415277640391194799202494066485063, 475874162103534761109503653193298906141757188998098147724536586722518146173], 'aut_phi_ratio': 160.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 4, 1], [2, 4, 2, 1], [2, 5, 1, 4], [2, 5, 4, 1], [2, 20, 2, 1], [4, 2, 4, 1], [4, 4, 2, 1], [4, 10, 4, 5], [4, 20, 2, 5], [5, 4, 1, 1], [10, 4, 1, 3], [10, 4, 4, 1], [10, 8, 4, 1], [20, 8, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_5:(C_4\\times C_2^8.C_2^3)', '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': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 4, 2], [2, 5, 8], [2, 20, 2], [4, 2, 4], [4, 4, 2], [4, 10, 20], [4, 20, 10], [5, 4, 1], [10, 4, 7], [10, 8, 4], [20, 8, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '80.50', 'commutator_count': 1, 'commutator_label': '20.5', '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', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 19151, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['320.1038', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 4, 1, 2], [2, 5, 1, 8], [2, 20, 1, 2], [4, 2, 2, 2], [4, 4, 1, 2], [4, 10, 2, 10], [4, 20, 1, 2], [4, 20, 2, 4], [5, 4, 1, 1], [10, 4, 1, 7], [10, 8, 2, 2], [20, 8, 2, 4]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 1249920, 'exponent': 20, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '160.236', 'hash': 19151, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [1, 2, 1, 4, 10], 'inner_gens': [[64801, 576071, 1984831, 1504171, 112891], [64801, 576071, 1984831, 224191, 2001541], [64801, 576071, 1984831, 1504171, 112891], [64801, 1856851, 1984831, 1504171, 112251], [64801, 2497401, 1984831, 1504811, 112891]], 'inner_hash': 50, 'inner_nilpotent': False, 'inner_order': 80, 'inner_split': False, 'inner_tex': 'C_2^2\\times F_5', 'inner_used': [2, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [4, 16], [8, 4]], 'label': '640.19151', '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': 'D10.(C4*D4)', 'ngens': 8, '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': 32, 'number_characteristic_subgroups': 54, 'number_conjugacy_classes': 76, 'number_divisions': 54, 'number_normal_subgroups': 226, 'number_subgroup_autclasses': 378, 'number_subgroup_classes': 928, 'number_subgroups': 4688, 'old_label': None, 'order': 640, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 95], [4, 416], [5, 4], [10, 60], [20, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 2, 4, 4, 2, 4], 'outer_gen_pows': [64001, 576001, 64961, 64001, 576641, 64001], 'outer_gens': [[1984031, 2497081, 1984831, 1505131, 2001381], [64801, 2496281, 1984831, 224351, 2001381], [704011, 1856531, 1984831, 865281, 720081], [704011, 576231, 1984831, 224831, 112251], [64801, 1216461, 1984831, 864161, 2000261], [1344821, 577031, 1984831, 1504811, 2001061]], 'outer_group': '512.752493', 'outer_hash': 7064241158567032655, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 64, 'outer_perms': [113385620764050802804782396285067162741423442538282616704425570297622113009630783225300137, 39752678459336177172150343515184337514444191040283920917006870808238230516356051276292261, 31097343862239644220695170975588800269971247221626124145187476750758630602601519339405740, 83699688203899273525730994533414787676497905588509753998420127555078205660038259896872409, 33818556096251790965136806067670261354999154329636491266385493997836660572026916710698899, 86967233660349148774030013497618780847493799603168552626907236199004572390329447950021981], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4.C_2^5', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 16], [8, 4], [16, 2]], 'representations': {'PC': {'code': '31967985302789130607066005006672587434609250664451', 'gens': [1, 2, 3, 4, 6], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 5, 5259, 91, 7021, 2525, 901, 141, 16142, 1374, 2054, 166, 16399, 3103, 2087]}, 'GLZN': {'d': 2, 'p': 40, 'gens': [113211, 64801, 113411, 2000421, 1344021, 576001, 64321, 480491]}, 'Perm': {'d': 15, 'gens': [7185388445, 12494321289, 368049, 374416, 16, 11536, 19678982400, 93924230400]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{10}.(C_4\\times D_4)', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 10], 'aut_gens': [[1, 4], [1, 12], [13, 4]], 'aut_group': '40.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 40, 'aut_permdeg': 7, 'aut_perms': [169, 2929], 'aut_phi_ratio': 2.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [4, 5, 2, 2], [5, 4, 1, 1], [10, 4, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_5', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 2], [4, 5, 4], [5, 4, 1], [10, 4, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '20.3', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['20.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [4, 5, 2, 2], [5, 4, 1, 1], [10, 4, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 20, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '40.12', 'hash': 12, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 5], 'inner_gens': [[1, 12], [33, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'F_5', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [4, 2]], 'label': '40.12', 'linC_count': 1, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*F5', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 10, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 16, 'number_subgroups': 40, 'old_label': None, 'order': 40, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 11], [4, 20], [5, 4], [10, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[21, 4]], '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': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 2]], 'representations': {'PC': {'code': 41624489227523, 'gens': [1, 3], 'pres': [4, -2, -2, -2, -5, 8, 146, 222, 34, 387, 263]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [975041, 15139867]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 127, 504]}, 'Perm': {'d': 7, 'gens': [151, 1, 288, 888]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times F_5', 'transitive_degree': 10, 'wreath_data': None, 'wreath_product': False}
