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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '3440.b', 'ambient_counter': 2, 'ambient_order': 3440, 'ambient_tex': 'Q_{16}\\times C_{215}', 'central': False, 'central_factor': True, 'centralizer_order': 430, 'characteristic': True, 'core_order': 16, 'counter': 26, 'cyclic': False, 'direct': True, 'hall': 2, 'label': '3440.b.215.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '215.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '215.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 215, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{215}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '16.9', 'subgroup_hash': 9, 'subgroup_order': 16, 'subgroup_tex': 'Q_{16}', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '3440.b', 'aut_centralizer_order': 168, 'aut_label': '215.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '32.43', 'aut_weyl_index': 168, 'centralizer': '8.a1.a1', 'complements': ['16.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['5.a1.a1', '43.a1.a1'], 'contains': ['430.a1.a1', '430.a1.b1', '430.b1.a1'], 'core': '215.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2366, 1696, 8083, 9125, 9028, 8888, 9692, 9629], 'generators': [617, 430], 'label': '3440.b.215.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '215.a1.a1', 'normal_contained_in': ['5.a1.a1', '43.a1.a1'], 'normal_contains': ['430.a1.b1', '430.a1.a1', '430.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '215.a1.a1', 'projective_image': '1720.30', 'quotient_action_image': '1.1', 'quotient_action_kernel': '215.1', 'quotient_action_kernel_order': 215, 'quotient_fusion': None, 'short_label': '215.a1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 8], 'aut_gens': [[1, 2], [1, 10], [1, 14], [3, 2]], 'aut_group': '32.43', 'aut_hash': 43, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [10824, 18366, 7679], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 4, 2, 1], [8, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_8:C_2', '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': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 4, 2], [8, 2, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '8.3', 'commutator_count': 1, 'commutator_label': '4.1', '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': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 4, 1, 2], [8, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 8, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, -1, 2]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '4.2', 'hash': 9, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4], 'inner_gens': [[1, 14], [5, 2]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 3]], 'label': '16.9', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'Q16', 'ngens': 2, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 7, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 7, 'number_subgroup_classes': 9, 'number_subgroups': 11, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 10], [8, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 2], 'outer_gens': [[1, 10], [3, 2]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1]], 'representations': {'PC': {'code': 39993887, 'gens': [1, 2], 'pres': [4, -2, 2, -2, -2, 32, 113, 21, 146, 34]}, 'Lie': [{'d': 2, 'q': 9, 'gens': [5109, 720, 1460, 3648], 'family': 'CSOPlus'}, {'d': 2, 'q': 7, 'gens': [2125, 2064, 1452, 675], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 7, 'gens': [1690, 2042]}, 'Perm': {'d': 16, 'gens': [12526762711680, 6920136155398, 4097467152142, 1313941673647]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_{16}', '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': '860.11', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [8, 2, 2, 4, 4, 42], 'aut_gens': [[1, 2], [431, 2], [1, 1722], [1, 862], [861, 2], [2065, 2754], [1521, 2242]], 'aut_group': None, 'aut_hash': 6966770769406336981, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 5376, 'aut_permdeg': 54, 'aut_perms': [52769072665096303387109426787547136011882552271857479251513755705104, 63626903938118612406400311733145594518625402971497667564592834213904, 10888866326238502035092606625280179658947014350142928246169958631264, 10888866326328945137901382107147898439228124955368049380082715506704, 836391475206143092662860288, 85824697030232801826567683209218642534089914175799201054574167204306541], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 4, 2, 1], [5, 1, 4, 1], [8, 2, 2, 1], [10, 1, 4, 1], [20, 2, 4, 1], [20, 4, 8, 1], [40, 2, 8, 1], [43, 1, 42, 1], [86, 1, 42, 1], [172, 2, 42, 1], [172, 4, 84, 1], [215, 1, 168, 1], [344, 2, 84, 1], [430, 1, 168, 1], [860, 2, 168, 1], [860, 4, 336, 1], [1720, 2, 336, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{84}\\times C_8:C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '672.1248', 'autcent_hash': 1248, 'autcent_nilpotent': True, 'autcent_order': 672, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_{84}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 4, 2], [5, 1, 4], [8, 2, 2], [10, 1, 4], [20, 2, 4], [20, 4, 8], [40, 2, 8], [43, 1, 42], [86, 1, 42], [172, 2, 42], [172, 4, 84], [215, 1, 168], [344, 2, 84], [430, 1, 168], [860, 2, 168], [860, 4, 336], [1720, 2, 336]], 'center_label': '430.4', 'center_order': 430, 'central_product': True, 'central_quotient': '8.3', 'commutator_count': 1, 'commutator_label': '4.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '43.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.9', 1], ['43.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 4, 1, 2], [5, 1, 4, 1], [8, 2, 2, 1], [10, 1, 4, 1], [20, 2, 4, 1], [20, 4, 4, 2], [40, 2, 8, 1], [43, 1, 42, 1], [86, 1, 42, 1], [172, 2, 42, 1], [172, 4, 42, 2], [215, 1, 168, 1], [344, 2, 84, 1], [430, 1, 168, 1], [860, 2, 168, 1], [860, 4, 168, 2], [1720, 2, 336, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 792, 'exponent': 1720, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 5, 43], 'faithful_reps': [[2, 0, 336]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '860.11', 'hash': 3120105997930144388, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4], 'inner_gens': [[1, 862], [2581, 2]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 672, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 860], [2, 645]], 'label': '3440.b', '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': True, 'monomial': True, 'name': 'Q16*C215', 'ngens': 6, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 20, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 1505, 'number_divisions': 24, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 36, 'number_subgroups': 44, 'old_label': None, 'order': 3440, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [4, 10], [5, 4], [8, 4], [10, 4], [20, 40], [40, 16], [43, 42], [86, 42], [172, 420], [215, 168], [344, 168], [430, 168], [860, 1680], [1720, 672]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [2, 2, 2, 84], 'outer_gen_pows': [3438, 0, 4, 4], 'outer_gens': [[431, 2], [1, 1722], [1061, 1202], [2295, 3274]], 'outer_group': '672.1248', 'outer_hash': 1248, 'outer_nilpotent': True, 'outer_order': 672, 'outer_permdeg': 20, 'outer_perms': [121645100408832000, 355687428096000, 1321086412800, 122002114192762473], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{84}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 64, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5, 43], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 4], [2, 1], [4, 5], [8, 1], [16, 1], [42, 4], [84, 1], [168, 5], [336, 1], [672, 1]], 'representations': {'PC': {'code': '3073331884335326122392911466231535576679', 'gens': [1, 2], 'pres': [6, -2, -2, -2, -2, -5, -43, 2928, 10345, 31, 31034, 50, 69, 178]}, 'GLFp': {'d': 2, 'p': 431, 'gens': [21161365840, 13999142305]}, 'Perm': {'d': 64, 'gens': [29741155484466632983345606526707847957462555704403391185425049059816142444202520420940313, 20203789539953222996300268551819550942349792604873931023641284766265158538593566720000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 430], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_{16}\\times C_{215}', 'transitive_degree': 3440, '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': '215.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 84, 'aut_gen_orders': [2, 84], 'aut_gens': [[1], [171], [212]], 'aut_group': '168.39', 'aut_hash': 39, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 168, 'aut_permdeg': 16, 'aut_perms': [1321086412800, 6746343273], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [43, 1, 42, 1], [215, 1, 168, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{84}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.39', 'autcent_hash': 39, 'autcent_nilpotent': True, 'autcent_order': 168, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{84}', '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], [5, 1, 4], [43, 1, 42], [215, 1, 168]], 'center_label': '215.1', 'center_order': 215, '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': ['5.1', '43.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['43.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [43, 1, 42, 1], [215, 1, 168, 1]], 'element_repr_type': 'PC', 'elementary': 215, 'eulerian_function': 1, 'exponent': 215, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [5, 43], 'faithful_reps': [[1, 0, 168]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '215.1', 'hash': 1, 'hyperelementary': 215, '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': 168, 'irrQ_dim': 168, 'irrR_degree': 2, 'irrep_stats': [[1, 215]], 'label': '215.1', 'linC_count': 168, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 46, 'linQ_degree_count': 1, 'linQ_dim': 46, 'linQ_dim_count': 1, 'linR_count': 84, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C215', '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': 215, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 215, 'order_factorization_type': 11, 'order_stats': [[1, 1], [5, 4], [43, 42], [215, 168]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [2, 84], 'outer_gen_pows': [0, 0], 'outer_gens': [[171], [212]], 'outer_group': '168.39', 'outer_hash': 39, 'outer_nilpotent': True, 'outer_order': 168, 'outer_permdeg': 16, 'outer_perms': [1321086412800, 6746343273], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{84}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 48, 'pgroup': 0, 'primary_abelian_invariants': [5, 43], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [42, 1], [168, 1]], 'representations': {'PC': {'code': 8987, 'gens': [1], 'pres': [2, -5, -43, 10]}, 'GLFp': {'d': 2, 'p': 431, 'gens': [10248063175]}, 'Perm': {'d': 48, 'gens': [1034492966044672722571857420614447919876790529556480000000000, 59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [215], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{215}', 'transitive_degree': 215, 'wreath_data': None, 'wreath_product': False}
