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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1760.415', 'ambient_counter': 415, 'ambient_order': 1760, 'ambient_tex': 'C_{11}:C_{10}\\times Q_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 44, 'characteristic': False, 'core_order': 176, 'counter': 28, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1760.415.10.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '10.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '10.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 10, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{10}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '176.39', 'subgroup_hash': 39, 'subgroup_order': 176, 'subgroup_tex': 'Q_8\\times C_{22}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.415', 'aut_centralizer_order': 44, 'aut_label': '10.a1', 'aut_quo_index': 4, 'aut_stab_index': 2, 'aut_weyl_group': '320.1632', 'aut_weyl_index': 88, 'centralizer': '40.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '5.a1.a1'], 'contains': ['20.a1.a1', '20.b1.a1', '20.b1.c1', '20.d1.a1', '20.e1.a1', '110.a1.a1'], 'core': '10.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5589, 2338, 4758, 4980, 4698, 5823, 3287, 5735], 'generators': [1, 440, 160, 10, 880], 'label': '1760.415.10.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '10.a1.a1', 'normal_contained_in': ['2.a1.a1', '5.a1.a1'], 'normal_contains': ['20.a1.a1', '20.b1.a1', '20.b1.c1', '110.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '10.a1.a1', 'projective_image': '440.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '10.a1.a1', 'subgroup_fusion': None, 'weyl_group': '40.10'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '88.12', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [2, 2, 2, 5, 3, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [1, 45, 139], [89, 2, 92], [1, 2, 84], [1, 2, 148], [1, 135, 51], [1, 3, 4], [1, 2, 5], [1, 2, 92], [1, 90, 92]], 'aut_group': '1920.236808', 'aut_hash': 1093768182639530347, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1920, 'aut_permdeg': 15, 'aut_perms': [88785889920, 87665276160, 1, 1728, 2003500800, 295076597760, 388630730880, 274471585920, 382403669760], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 6, 1], [11, 1, 10, 1], [22, 1, 10, 1], [22, 1, 20, 1], [44, 2, 60, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3:S_4\\times C_{10}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '320.1523', 'autcent_hash': 1523, 'autcent_nilpotent': True, 'autcent_order': 320, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5:C_{10}', '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, 3], [4, 2, 6], [11, 1, 10], [22, 1, 30], [44, 2, 60]], 'center_label': '44.4', 'center_order': 44, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 39, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['2.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 6], [11, 1, 10, 1], [22, 1, 10, 3], [44, 2, 10, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 931, 'exponent': 44, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '88.12', 'hash': 39, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2], 'inner_gens': [[1, 2, 4], [1, 2, 92], [1, 90, 4]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 88], [2, 22]], 'label': '176.39', 'linC_count': 960, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 8, 'linQ_dim': 14, 'linQ_dim_count': 8, 'linR_count': 40, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'Q8*C22', '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 110, 'number_divisions': 20, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 38, 'number_subgroups': 38, 'old_label': None, 'order': 176, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 12], [11, 10], [22, 30], [44, 120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [2, 6, 2, 2, 10], 'outer_gen_pows': [4, 0, 0, 0, 0], 'outer_gens': [[1, 46, 4], [89, 134, 138], [1, 3, 5], [1, 2, 84], [1, 3, 36]], 'outer_group': '480.1198', 'outer_hash': 1198, 'outer_nilpotent': False, 'outer_order': 480, 'outer_permdeg': 13, 'outer_perms': [3628807, 43545607, 1520467200, 16, 482676600], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times C_{10}\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 11], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [10, 8], [20, 2]], 'representations': {'PC': {'code': 142968986036600009, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -11, 446, 697, 42, 58]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [826574, 359380, 60644, 36037, 48302]}, 'Perm': {'d': 21, 'gens': [2702889776069683200, 5245011362810419200, 479001600, 36288000, 7817361150241843200]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 22], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_8\\times C_{22}', 'transitive_degree': 176, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 440, 'aut_gen_orders': [40, 20, 20, 10, 40, 22], 'aut_gens': [[1, 2, 20], [881, 1663, 500], [1, 762, 341], [1, 1463, 700], [1, 243, 700], [881, 1143, 501], [881, 362, 460]], 'aut_group': None, 'aut_hash': 2016333457522889885, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 28160, 'aut_permdeg': 27, 'aut_perms': [10369587246467483896247349409, 4270639055280423088250262837, 6165721406643483447285903212, 8465483255827783683968956205, 2370819651866855753427559784, 4484037795055987019684416173], 'aut_phi_ratio': 44.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 1, 2], [4, 4, 4, 1], [5, 11, 1, 4], [8, 2, 4, 1], [10, 11, 1, 4], [10, 11, 2, 4], [11, 5, 2, 1], [20, 22, 1, 8], [20, 44, 4, 4], [22, 5, 2, 1], [22, 5, 4, 1], [40, 22, 4, 4], [44, 10, 2, 2], [44, 20, 8, 1], [88, 10, 8, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2\\times C_{44}).C_{10}.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 220, 'autcentquo_group': '880.118', 'autcentquo_hash': 118, 'autcentquo_nilpotent': False, 'autcentquo_order': 880, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 4, 4], [5, 11, 4], [8, 2, 4], [10, 11, 12], [11, 5, 2], [20, 22, 8], [20, 44, 16], [22, 5, 6], [40, 22, 16], [44, 10, 4], [44, 20, 8], [88, 10, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '440.13', 'commutator_count': 1, 'commutator_label': '44.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 415, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.9', 1], ['2.1', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 4, 1, 4], [5, 11, 4, 1], [8, 2, 2, 2], [10, 11, 4, 3], [11, 5, 2, 1], [20, 22, 4, 2], [20, 44, 4, 4], [22, 5, 2, 3], [40, 22, 8, 2], [44, 10, 2, 2], [44, 20, 2, 4], [88, 10, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 62496, 'exponent': 440, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '440.44', 'hash': 415, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 220, 'inner_gen_orders': [1, 10, 44], 'inner_gens': [[1, 2, 20], [1, 2, 300], [1, 1482, 20]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 440, 'inner_split': False, 'inner_tex': 'C_{44}:C_{10}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 30], [5, 16], [10, 12]], 'label': '1760.415', 'linC_count': 160, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 8, 'linQ_dim': 18, 'linQ_dim_count': 8, 'linR_count': 48, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C11:C10*Q16', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 42, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 98, 'number_divisions': 36, 'number_normal_subgroups': 66, 'number_subgroup_autclasses': 64, 'number_subgroup_classes': 120, 'number_subgroups': 532, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [4, 20], [5, 44], [8, 8], [10, 132], [11, 10], [20, 880], [22, 30], [40, 352], [44, 200], [88, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [220, 1540, 0, 440, 440], 'outer_gens': [[881, 223, 420], [1, 223, 900], [1, 882, 1300], [1, 1323, 900], [1, 1103, 901]], 'outer_group': '64.202', 'outer_hash': 202, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [5160, 126, 127, 367920, 806542], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 29, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 10], [8, 2], [10, 8], [16, 2], [20, 2], [40, 2]], 'representations': {'PC': {'code': 4744006602381113073940507746820810725605727191660030009, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -5, -2, -2, -2, -11, 36, 4636, 4210, 6877, 80, 10511, 1768, 102, 25212, 4219, 124, 15693, 9820]}, 'GLZN': {'d': 2, 'p': 77, 'gens': [20872622, 29218127, 457073, 30983305, 5807055, 19630962, 15522156]}, 'Perm': {'d': 29, 'gens': [1, 427033839765991, 799819538072784, 11728439597242967934246912000, 1181150107543566, 1559624222783880, 327521760645639334802227200000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}:C_{10}\\times Q_{16}', 'transitive_degree': 352, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [7]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', '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], [5, 1, 4], [10, 1, 4]], 'center_label': '10.2', 'center_order': 10, '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', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 10, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 10]], 'label': '10.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C10', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 10, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 10, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [5, 4], [10, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[7]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2]], 'representations': {'PC': {'code': 83, 'gens': [1], 'pres': [2, -2, -5, 4]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16717348]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504]}, 'Perm': {'d': 7, 'gens': [720, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}', 'transitive_degree': 10, 'wreath_data': None, 'wreath_product': False}
