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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '960.10067', 'ambient_counter': 10067, 'ambient_order': 960, 'ambient_tex': 'C_2^2:D_{12}\\times C_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 240, 'characteristic': False, 'core_order': 60, 'counter': 47, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.10067.8.r1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.r1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 8, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '120.31', 'subgroup_hash': 31, 'subgroup_order': 120, 'subgroup_tex': 'C_2\\times C_{60}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.10067', 'aut_centralizer_order': 192, 'aut_label': '8.r1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '64.196', 'aut_weyl_index': 1536, 'centralizer': '4.d1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['4.d1', '4.n1'], 'contains': ['16.d1', '16.o1', '24.q1', '40.r1'], 'core': '16.d1', 'coset_action_label': None, 'count': 8, 'diagramx': [237, -1, 215, -1], 'generators': [1832445, 2501513, 2129625, 700098, 3706515], 'label': '960.10067.8.r1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.d1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.d1', 'old_label': '8.r1', 'projective_image': '48.38', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.r1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '120.31', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 4, 4], 'aut_gens': [[1, 2], [1, 82], [1, 99], [1, 74], [61, 63]], 'aut_group': '64.196', 'aut_hash': 196, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 10, 'aut_perms': [120, 806526, 408367, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 2, 1], [4, 1, 4, 1], [5, 1, 4, 1], [6, 1, 2, 1], [6, 1, 4, 1], [10, 1, 4, 1], [10, 1, 8, 1], [12, 1, 8, 1], [15, 1, 8, 1], [20, 1, 16, 1], [30, 1, 8, 1], [30, 1, 16, 1], [60, 1, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4^2:C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.196', 'autcent_hash': 196, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4^2:C_2^2', '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, 3], [3, 1, 2], [4, 1, 4], [5, 1, 4], [6, 1, 6], [10, 1, 12], [12, 1, 8], [15, 1, 8], [20, 1, 16], [30, 1, 24], [60, 1, 32]], 'center_label': '120.31', 'center_order': 120, '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', '3.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['4.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [4, 1, 2, 2], [5, 1, 4, 1], [6, 1, 2, 3], [10, 1, 4, 3], [12, 1, 4, 2], [15, 1, 8, 1], [20, 1, 8, 2], [30, 1, 8, 3], [60, 1, 16, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 72, 'exponent': 60, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '60.13', 'hash': 31, '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, 120]], 'label': '120.31', 'linC_count': 2304, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 28, 'linQ_dim': 8, 'linQ_dim_count': 28, 'linR_count': 32, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C60', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, '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': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 120, 'number_divisions': 24, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 32, 'number_subgroups': 32, 'old_label': None, 'order': 120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [5, 4], [6, 6], [10, 12], [12, 8], [15, 8], [20, 16], [30, 24], [60, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 82], [1, 99], [1, 74], [61, 63]], 'outer_group': '64.196', 'outer_hash': 196, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [120, 806526, 408367, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2:C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 6], [8, 6], [16, 2]], 'representations': {'PC': {'code': 251909912583, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -3, -5, 26, 42, 78]}, 'GLFp': {'d': 2, 'p': 61, 'gens': [10441130, 2496792]}, 'Perm': {'d': 14, 'gens': [127008000, 6227020800, 10080, 96, 40279680]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 60], '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_{60}', 'transitive_degree': 120, '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': '80.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [12, 4, 4, 4, 12, 12, 4], 'aut_gens': [[86547, 86153, 1832445, 128745, 247176], [3207607, 3663923, 1788885, 128745, 889226], [3525195, 2001847, 1959255, 128745, 3147574], [1833063, 127777, 2002815, 128745, 565028], [2501679, 127777, 1832445, 128745, 2439076], [3206639, 86153, 1959255, 128745, 932764], [268441, 3663923, 2002815, 128745, 1435614], [1290989, 1960223, 1959255, 128745, 2395494]], 'aut_group': None, 'aut_hash': 4741775015152245913, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 98304, 'aut_permdeg': 44, 'aut_perms': [1004393210783299279408336874317724936179284107181255188, 50047040516485814017843489427125074350402054226874136, 2380178340380274066439735642696738319745428928462524153, 1478252567638813504004164309053489697079888568432320596, 1191494951124637646766467825399589726838324039951886412, 239185770334432054198250281317461199483074297515366332, 2415037377764582737363145724762313198951208910634270130], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 1, 4, 1], [2, 2, 4, 1], [2, 6, 8, 1], [2, 12, 2, 1], [3, 2, 1, 1], [4, 4, 4, 1], [4, 12, 2, 1], [5, 1, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [6, 4, 4, 1], [10, 1, 4, 1], [10, 1, 8, 1], [10, 1, 16, 1], [10, 2, 16, 1], [10, 6, 32, 1], [10, 12, 8, 1], [12, 4, 8, 1], [15, 2, 4, 1], [20, 4, 16, 1], [20, 12, 8, 1], [30, 2, 4, 1], [30, 2, 8, 1], [30, 2, 16, 1], [30, 4, 16, 1], [60, 4, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^6.C_2^5.C_2^4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 7903657167058653875, 'autcent_nilpotent': True, 'autcent_order': 8192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7.C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 4], [2, 6, 8], [2, 12, 2], [3, 2, 1], [4, 4, 4], [4, 12, 2], [5, 1, 4], [6, 2, 7], [6, 4, 4], [10, 1, 28], [10, 2, 16], [10, 6, 32], [10, 12, 8], [12, 4, 8], [15, 2, 4], [20, 4, 16], [20, 12, 8], [30, 2, 28], [30, 4, 16], [60, 4, 32]], 'center_label': '40.14', 'center_order': 40, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '12.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10067, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1], ['96.89', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 4], [2, 6, 1, 8], [2, 12, 1, 2], [3, 2, 1, 1], [4, 4, 1, 4], [4, 12, 1, 2], [5, 1, 4, 1], [6, 2, 1, 7], [6, 4, 1, 4], [10, 1, 4, 7], [10, 2, 4, 4], [10, 6, 4, 8], [10, 12, 4, 2], [12, 4, 2, 4], [15, 2, 4, 1], [20, 4, 4, 4], [20, 12, 4, 2], [30, 2, 4, 7], [30, 4, 4, 4], [60, 4, 8, 4]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 2555280, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '240.206', 'hash': 10067, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 1, 1, 6], 'inner_gens': [[86547, 86153, 1832445, 128745, 2395494], [86547, 86153, 1832445, 128745, 2121202], [86547, 86153, 1832445, 128745, 247176], [86547, 86153, 1832445, 128745, 247176], [1692793, 127777, 1832445, 128745, 247176]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': False, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 140], [4, 20]], 'label': '960.10067', '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': 'C2^2:D12*C10', '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, 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': 30, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 240, 'number_divisions': 88, 'number_normal_subgroups': 286, 'number_subgroup_autclasses': 256, 'number_subgroup_classes': 1324, 'number_subgroups': 4176, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 87], [3, 2], [4, 40], [5, 4], [6, 30], [10, 348], [12, 32], [15, 8], [20, 160], [30, 120], [60, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 8, 'outer_gen_orders': [4, 4, 4, 4, 4, 2], 'outer_gen_pows': [85185, 85185, 85185, 85185, 85185, 85185], 'outer_gens': [[85579, 2001847, 1788885, 128745, 2902474], [1832095, 86153, 1788885, 128745, 569158], [1833063, 2001847, 1788885, 128745, 1714058], [3664497, 3705547, 1959255, 128745, 2902474], [2002197, 1831477, 1959255, 128745, 1071986], [1789503, 2001847, 1832445, 128745, 708358]], 'outer_group': None, 'outer_hash': 6206307954038007969, 'outer_nilpotent': True, 'outer_order': 4096, 'outer_permdeg': 20, 'outer_perms': [392327156256928414, 628880218730668534, 6314124408, 891063873249608658, 499803785547909865, 732297646101744], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2.C_2^6.C_2^5', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 24], [8, 20], [16, 8]], 'representations': {'PC': {'code': 105771540236547054232178572888375303, 'gens': [1, 2, 3, 4, 5], 'pres': [8, -2, -2, -2, -2, 2, -2, -3, -5, 7044, 492, 116, 16901, 141, 39430, 222]}, 'GLZN': {'d': 2, 'p': 44, 'gens': [2501513, 3567787, 1959255, 766665, 700560, 1788885, 86153, 2002815]}, 'Perm': {'d': 18, 'gens': [21009971849040, 87660962640, 482721120, 88219757520, 33, 482630400, 41040, 376610217984000]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_2^2:D_{12}\\times C_{10}', 'transitive_degree': 240, 'wreath_data': None, 'wreath_product': False}
