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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1600.5593', 'ambient_counter': 5593, 'ambient_order': 1600, 'ambient_tex': 'C_{16}.C_{10}^2', 'central': False, 'central_factor': False, 'centralizer_order': 800, 'characteristic': False, 'core_order': 80, 'counter': 26, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1600.5593.20.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '20.e1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '20.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_{10}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '80.2', 'subgroup_hash': 2, 'subgroup_order': 80, 'subgroup_tex': 'C_{80}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1600.5593', 'aut_centralizer_order': 160, 'aut_label': '20.e1', 'aut_quo_index': 3, 'aut_stab_index': 18, 'aut_weyl_group': '32.21', 'aut_weyl_index': 2880, 'centralizer': '2.a1', 'complements': [], 'conjugacy_class_count': 18, 'contained_in': ['4.e1', '10.a1', '10.b1'], 'contains': ['40.c1', '100.e1'], 'core': '20.e1', 'coset_action_label': None, 'count': 18, 'diagramx': [1674, 8116, 8372, 1628], 'generators': [505, 1000, 400, 2, 800], 'label': '1600.5593.20.e1', 'mobius_quo': 0, 'mobius_sub': -2, 'normal_closure': '20.e1', 'normal_contained_in': ['4.e1', '10.a1', '10.b1'], 'normal_contains': ['40.c1', '100.e1'], 'normalizer': '1.a1', 'old_label': '20.e1', 'projective_image': '40.14', 'quotient_action_image': '2.1', 'quotient_action_kernel': '10.2', 'quotient_action_kernel_order': 10, 'quotient_fusion': None, 'short_label': '20.e1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '80.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [2, 4, 4], 'aut_gens': [[1], [71], [21], [17]], 'aut_group': '32.21', 'aut_hash': 21, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 10, 'aut_perms': [362880, 368760, 9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [8, 1, 4, 1], [10, 1, 4, 1], [16, 1, 8, 1], [20, 1, 8, 1], [40, 1, 16, 1], [80, 1, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_4^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.21', 'autcent_hash': 21, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4^2', '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], [4, 1, 2], [5, 1, 4], [8, 1, 4], [10, 1, 4], [16, 1, 8], [20, 1, 8], [40, 1, 16], [80, 1, 32]], 'center_label': '80.2', 'center_order': 80, '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', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['16.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [8, 1, 4, 1], [10, 1, 4, 1], [16, 1, 8, 1], [20, 1, 8, 1], [40, 1, 16, 1], [80, 1, 32, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 80, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 32]], 'familial': True, 'frattini_label': '8.1', 'frattini_quotient': '10.2', 'hash': 2, 'hyperelementary': 10, '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': 32, 'irrQ_dim': 32, 'irrR_degree': 2, 'irrep_stats': [[1, 80]], 'label': '80.2', 'linC_count': 32, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 2, 'linQ_dim': 12, 'linQ_dim_count': 2, 'linR_count': 16, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C80', 'ngens': 5, 'nilpotency_class': 1, '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 80, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 4], [8, 4], [10, 4], [16, 8], [20, 8], [40, 16], [80, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[71], [21], [17]], 'outer_group': '32.21', 'outer_hash': 21, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 368760, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4^2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [16, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 3], [8, 2], [16, 1], [32, 1]], 'representations': {'PC': {'code': 9140562836483, 'gens': [1], 'pres': [5, -2, -2, -2, -2, -5, 10, 26, 42, 58]}, 'GLFp': {'d': 2, 'p': 31, 'gens': [528784]}, 'Perm': {'d': 21, 'gens': [38277836852935680000, 96, 17787216907833246720, 7542061610582953440, 2439325392333218640]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [80], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{80}', 'transitive_degree': 80, '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': '800.948', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [24, 12, 60, 12, 4], 'aut_gens': [[1, 10, 100], [413, 61, 100], [1295, 874, 900], [1299, 809, 700], [830, 431, 300], [74, 865, 1300]], 'aut_group': None, 'aut_hash': 3409991336813292700, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 92160, 'aut_permdeg': 38, 'aut_perms': [76658849292193668558583040148913615370462398, 205643782935713064395171462090950948631905812, 173143763952213541670986646686219033186176333, 174353607913258668916439370002157886662972342, 64197782657223842085130264260047663818590100], 'aut_phi_ratio': 144.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [4, 1, 2, 1], [4, 2, 3, 1], [5, 1, 24, 1], [8, 1, 4, 1], [8, 2, 6, 1], [10, 1, 24, 1], [10, 2, 72, 1], [16, 1, 8, 1], [16, 2, 12, 1], [20, 1, 48, 1], [20, 2, 72, 1], [40, 1, 96, 1], [40, 2, 144, 1], [80, 1, 192, 1], [80, 2, 288, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2^2\\times A_4).C_2^4.S_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 120, 'autcent_group': None, 'autcent_hash': 4293128808008943807, 'autcent_nilpotent': False, 'autcent_order': 15360, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '(C_2^3\\times C_4^2).S_5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [4, 1, 2], [4, 2, 3], [5, 1, 24], [8, 1, 4], [8, 2, 6], [10, 1, 24], [10, 2, 72], [16, 1, 8], [16, 2, 12], [20, 1, 48], [20, 2, 72], [40, 1, 96], [40, 2, 144], [80, 1, 192], [80, 2, 288]], 'center_label': '400.51', 'center_order': 400, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 5593, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 2], ['64.185', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [4, 1, 2, 1], [4, 2, 1, 3], [5, 1, 4, 6], [8, 1, 4, 1], [8, 2, 2, 3], [10, 1, 4, 6], [10, 2, 4, 18], [16, 1, 8, 1], [16, 2, 4, 3], [20, 1, 8, 6], [20, 2, 4, 18], [40, 1, 16, 6], [40, 2, 8, 18], [80, 1, 32, 6], [80, 2, 16, 18]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 13888, 'exponent': 80, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '200.52', 'hash': 5593, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1], 'inner_gens': [[1, 810, 100], [801, 10, 100], [1, 10, 100]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 800], [2, 200]], 'label': '1600.5593', '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': 'C16.C10^2', 'ngens': 8, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 18, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 1000, 'number_divisions': 119, 'number_normal_subgroups': 312, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 336, 'number_subgroups': 360, 'old_label': None, 'order': 1600, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 7], [4, 8], [5, 24], [8, 16], [10, 168], [16, 32], [20, 192], [40, 384], [80, 768]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 120, 'outer_gen_orders': [4, 12, 12, 6, 4], 'outer_gen_pows': [0, 0, 0, 1, 0], 'outer_gens': [[856, 885, 900], [894, 1253, 1100], [894, 1237, 700], [63, 417, 900], [3, 479, 1300]], 'outer_group': None, 'outer_hash': 1256762992678057555, 'outer_nilpotent': False, 'outer_order': 23040, 'outer_permdeg': 33, 'outer_perms': [1638820610548691462289632293133765785, 2459738607777952158449890184486400003, 1910480418080372963498197249357722747, 4617543574037884825504252603725095549, 559278952341043850779881814917240965], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '(C_2\\times C_4^2\\times S_3).S_5', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 42, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 8, 5, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 52], [8, 24], [16, 25], [64, 6]], 'representations': {'PC': {'code': 23626608279467703874661911966464, 'gens': [1, 3, 5], 'pres': [8, -2, -5, -2, -5, -2, -2, -2, -2, 16, 19442, 66, 116, 141, 166]}, 'Perm': {'d': 42, 'gens': [787776892770397050131677661048970189138296832000000, 342872325536638508718198878433162125790935350041600, 274315015579881341349264821001458277060146626560000, 239146861610491088301865156883687891228572368076800, 101989452441259720293704396075126623631135431449600, 33472938269339935293570026045548616721747392313600, 1451520, 96]}}, 'schur_multiplier': [2, 10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10, 40], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{16}.C_{10}^2', 'transitive_degree': 800, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1, 2], [10, 13], [10, 15]], 'aut_group': '24.5', 'aut_hash': 5, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [127, 857], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 4, 1], [10, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.5', 'autcent_hash': 5, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times S_3', '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, 3], [5, 1, 4], [10, 1, 12]], 'center_label': '20.5', 'center_order': 20, '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', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 1], [10, 1, 4, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '20.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 20]], 'label': '20.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 6, 'linQ_dim': 5, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C10', '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 20, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [5, 4], [10, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[10, 13], [10, 15]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [127, 857], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4]], 'representations': {'PC': {'code': 2179, 'gens': [1, 2], 'pres': [3, -2, -2, -5, 16]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706303489327]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [13320, 4001]}, 'Perm': {'d': 9, 'gens': [40320, 720, 96]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{10}', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
