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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '288.642', 'ambient_counter': 642, 'ambient_order': 288, 'ambient_tex': 'C_{12}^2:C_2', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 16, 'counter': 22, 'cyclic': False, 'direct': None, 'hall': 2, 'label': '288.642.9.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '9.a1', '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': 9, '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': '32.21', 'subgroup_hash': 21, 'subgroup_order': 32, 'subgroup_tex': 'C_2\\times C_4^2', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '288.642', 'aut_centralizer_order': 4, 'aut_label': '9.a1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '384.20100', 'aut_weyl_index': 12, 'centralizer': '3.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1', '3.b1'], 'contains': ['18.a1', '18.b1', '18.c1'], 'core': '18.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [1524, -1, 7854, -1], 'generators': [1, 18, 216], 'label': '288.642.9.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '3.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1', 'old_label': '9.a1', 'projective_image': '18.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '9.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '32.21', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 3, 2, 2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 8], [1, 30, 8], [1, 28, 14], [1, 3, 9], [1, 3, 8], [21, 2, 8], [17, 2, 8], [1, 22, 8], [1, 18, 8], [1, 6, 28], [1, 22, 24]], 'aut_group': '1536.408569020', 'aut_hash': 2246280721724691594, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 24, 'aut_perms': [26570953763857078535902, 216534374421977646116524, 539649757487563306680528, 1026578433973640226904, 132787916831906967859200, 26217569221913663596800, 104618556585332979961, 360693186001987547753, 132992765108218816624669, 26166479491951951783798], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 1, 4, 1], [4, 1, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '1536.408569020', 'autcent_hash': 2246280721724691594, 'autcent_nilpotent': False, 'autcent_order': 1536, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^6:S_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, 7], [4, 1, 24]], 'center_label': '32.21', 'center_order': 32, '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', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 21, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 1, 2, 12]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 7, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '8.5', 'hash': 21, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 8], [1, 2, 8], [1, 2, 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, 32]], 'label': '32.21', 'linC_count': 1792, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 192, 'linQ_dim': 5, 'linQ_dim_count': 192, 'linR_count': 192, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*C4^2', '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': 32, 'number_divisions': 20, 'number_normal_subgroups': 54, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 54, 'number_subgroups': 54, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 30, 8], [1, 28, 14], [1, 3, 9], [1, 3, 8], [21, 2, 8], [17, 2, 8], [1, 22, 8], [1, 18, 8], [1, 6, 28], [1, 22, 24]], 'outer_group': '1536.408569020', 'outer_hash': 2246280721724691594, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 24, 'outer_perms': [26570953763857078535902, 216534374421977646116524, 539649757487563306680528, 1026578433973640226904, 132787916831906967859200, 26217569221913663596800, 104618556585332979961, 360693186001987547753, 132992765108218816624669, 26166479491951951783798], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 4, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12]], 'representations': {'PC': {'code': 589834, 'gens': [1, 2, 4], 'pres': [5, 2, 2, 2, 2, 2, 26, 58]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [705946296092, 141602720900, 706201387282]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [372730, 1821544, 1281853, 1402363, 1565004]}, 'Perm': {'d': 10, 'gens': [16560, 22, 362880, 5160, 7]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 4], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4^2', 'transitive_degree': 32, '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': '96.161', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [4, 6, 6, 12, 4], 'aut_gens': [[1, 2, 24], [145, 80, 246], [145, 218, 198], [49, 80, 138], [253, 82, 168], [1, 236, 102]], 'aut_group': None, 'aut_hash': 5086249565183209125, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4608, 'aut_permdeg': 26, 'aut_perms': [42766752908076940174699128, 40288613454356838022787198, 322279479489402192649118190, 90059014965659216530563962, 4868466125886558210693048], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 4, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 12, 1], [4, 3, 12, 1], [6, 1, 6, 1], [6, 2, 3, 1], [6, 2, 6, 1], [6, 3, 8, 1], [12, 1, 24, 1], [12, 2, 12, 1], [12, 2, 24, 1], [12, 3, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_2^4:C_3.D_4\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '768.1090187', 'autcent_hash': 1090187, 'autcent_nilpotent': False, 'autcent_order': 768, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3:\\GL(2,\\mathbb{Z}/4)', '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], [2, 3, 4], [3, 1, 2], [3, 2, 3], [4, 1, 12], [4, 3, 12], [6, 1, 6], [6, 2, 9], [6, 3, 8], [12, 1, 24], [12, 2, 36], [12, 3, 24]], 'center_label': '48.20', 'center_order': 48, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 642, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 2], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 6], [4, 3, 2, 6], [6, 1, 2, 3], [6, 2, 1, 3], [6, 2, 2, 3], [6, 3, 2, 4], [12, 1, 4, 6], [12, 2, 2, 6], [12, 2, 4, 6], [12, 3, 4, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1456, 'exponent': 12, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '72.48', 'hash': 642, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 1, 3], 'inner_gens': [[1, 2, 120], [1, 2, 24], [193, 2, 24]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 96], [2, 48]], 'label': '288.642', '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': 'C12^2:C2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 144, 'number_divisions': 60, 'number_normal_subgroups': 138, 'number_subgroup_autclasses': 66, 'number_subgroup_classes': 231, 'number_subgroups': 402, 'old_label': None, 'order': 288, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 15], [3, 8], [4, 48], [6, 48], [12, 168]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2, 2, 2, 4, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 158, 36], [157, 154, 168], [1, 88, 174], [1, 22, 24], [145, 14, 24], [145, 154, 24], [13, 166, 36], [1, 86, 180], [157, 146, 168]], 'outer_group': '768.1090187', 'outer_hash': 1090187, 'outer_nilpotent': False, 'outer_order': 768, 'outer_permdeg': 14, 'outer_perms': [7, 1174326, 19160789766, 13413219126, 806407, 19680156847, 6268111926, 39492271134, 1174327], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:\\GL(2,\\mathbb{Z}/4)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 24], [4, 22], [8, 6]], 'representations': {'PC': {'code': 9871553586395837595327241, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -3, -2, -2, -3, 36, 58, 4204, 102, 10085, 124, 9414]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [2356776259, 3503869336, 3615679239, 9732836327, 9789111396, 6526074264, 7487656956]}, 'Perm': {'d': 14, 'gens': [6314112000, 9, 5889, 13971242880, 11536, 16, 20122058880]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}^2:C_2', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
