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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1248.959', 'ambient_counter': 959, 'ambient_order': 1248, 'ambient_tex': 'C_4\\times S_3\\times C_{52}', 'central': False, 'central_factor': False, 'centralizer_order': 624, 'characteristic': False, 'core_order': 78, 'counter': 22, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1248.959.16.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '16.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '16.10', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 10, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2\\times C_4', 'simple': False, 'solvable': True, 'special_labels': ['C4'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '78.6', 'subgroup_hash': 6, 'subgroup_order': 78, 'subgroup_tex': 'C_{78}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1248.959', 'aut_centralizer_order': 384, 'aut_label': '16.a1', 'aut_quo_index': 12, 'aut_stab_index': 3, 'aut_weyl_group': '24.9', 'aut_weyl_index': 1152, 'centralizer': '2.c1', 'complements': [], 'conjugacy_class_count': 3, 'contained_in': ['8.a1', '8.b1', '8.c1', '8.d1'], 'contains': ['32.a1', '48.a1', '208.a1'], 'core': '16.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [7983, 3644, 1052, 3198], 'generators': [624, 96, 416], 'label': '1248.959.16.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '16.a1', 'normal_contained_in': ['8.a1', '8.b1', '8.c1', '8.d1'], 'normal_contains': ['32.a1', '48.a1', '208.a1'], 'normalizer': '1.a1', 'old_label': '16.a1', 'projective_image': '48.35', 'quotient_action_image': '2.1', 'quotient_action_kernel': '8.2', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '16.a1', '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': '78.6', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1], [53], [67]], 'aut_group': '24.9', 'aut_hash': 9, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 24, 'aut_permdeg': 9, 'aut_perms': [40320, 867], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1], [13, 1, 12, 1], [26, 1, 12, 1], [39, 1, 24, 1], [78, 1, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{12}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.9', 'autcent_hash': 9, 'autcent_nilpotent': True, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{12}', '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], [3, 1, 2], [6, 1, 2], [13, 1, 12], [26, 1, 12], [39, 1, 24], [78, 1, 24]], 'center_label': '78.6', 'center_order': 78, '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', '3.1', '13.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['13.1', 1], ['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1], [13, 1, 12, 1], [26, 1, 12, 1], [39, 1, 24, 1], [78, 1, 24, 1]], 'element_repr_type': 'PC', 'elementary': 78, 'eulerian_function': 1, 'exponent': 78, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 13], 'faithful_reps': [[1, 0, 24]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '78.6', 'hash': 6, 'hyperelementary': 78, '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': 24, 'irrQ_dim': 24, 'irrR_degree': 2, 'irrep_stats': [[1, 78]], 'label': '78.6', 'linC_count': 24, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 3, 'linQ_dim': 14, 'linQ_dim_count': 3, 'linR_count': 12, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C78', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 78, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 78, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2], [13, 12], [26, 12], [39, 24], [78, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[53], [67]], 'outer_group': '24.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [40320, 867], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{12}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 13], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [12, 2], [24, 2]], 'representations': {'PC': {'code': 5037143, 'gens': [1], 'pres': [3, -2, -3, -13, 6, 22]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [2211, 26376, 8792]}, 'Perm': {'d': 18, 'gens': [355687428096000, 2615348736000, 5748019200]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [78], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{78}', 'transitive_degree': 78, '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': '416.175', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 12, 12, 2, 12], 'aut_gens': [[1, 2, 8], [833, 2, 232], [629, 316, 570], [1, 2, 1166], [417, 6, 426], [833, 314, 8]], 'aut_group': None, 'aut_hash': 2040933745014014244, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 27648, 'aut_permdeg': 36, 'aut_perms': [186300950093979142167697521747536249557830, 234873953614246683492605082362669658209265, 9115428890653686012994657787758552842372, 103806457820159884346339462082719112022945, 187102108031941297639534745969335870703505], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 4, 1], [3, 2, 1, 1], [4, 1, 12, 1], [4, 3, 12, 1], [6, 2, 3, 1], [12, 2, 12, 1], [13, 1, 12, 1], [26, 1, 36, 1], [26, 3, 48, 1], [39, 2, 12, 1], [52, 1, 144, 1], [52, 3, 144, 1], [78, 2, 36, 1], [156, 2, 144, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{12}\\times C_2^4:C_3.D_4\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': None, 'autcent_hash': 7103775244321617999, 'autcent_nilpotent': False, 'autcent_order': 4608, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_{12}\\times C_2^4:C_3.D_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, 2, 1], [4, 1, 12], [4, 3, 12], [6, 2, 3], [12, 2, 12], [13, 1, 12], [26, 1, 36], [26, 3, 48], [39, 2, 12], [52, 1, 144], [52, 3, 144], [78, 2, 36], [156, 2, 144]], 'center_label': '208.20', 'center_order': 208, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '13.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 959, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['13.1', 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, 2, 1, 1], [4, 1, 2, 6], [4, 3, 2, 6], [6, 2, 1, 3], [12, 2, 2, 6], [13, 1, 12, 1], [26, 1, 12, 3], [26, 3, 12, 4], [39, 2, 12, 1], [52, 1, 24, 6], [52, 3, 24, 6], [78, 2, 12, 3], [156, 2, 24, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 20496, 'exponent': 156, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 13], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '312.59', 'hash': 959, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 1, 3], 'inner_gens': [[1, 2, 424], [1, 2, 8], [833, 2, 8]], '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': None, 'irrep_stats': [[1, 416], [2, 208]], 'label': '1248.959', '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': 'C4*S3*C52', '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': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 624, 'number_divisions': 60, 'number_normal_subgroups': 138, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 216, 'number_subgroups': 372, 'old_label': None, 'order': 1248, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 15], [3, 2], [4, 48], [6, 6], [12, 24], [13, 12], [26, 180], [39, 24], [52, 576], [78, 72], [156, 288]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [12, 6, 12, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[629, 318, 850], [629, 940, 494], [1, 314, 968], [1, 626, 590]], 'outer_group': None, 'outer_hash': 7103775244321617999, 'outer_nilpotent': False, 'outer_order': 4608, 'outer_permdeg': 19, 'outer_perms': [19918589506074139, 6423470892478923, 174476356558, 19918496056872237], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_{12}\\times C_2^4:C_3.D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 4, 13], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 6], [12, 8], [24, 16], [48, 6]], 'representations': {'PC': {'code': 309393186080692172554600588796461079, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -2, -2, -2, -3, -13, 36, 11875, 80, 29684, 102, 18821, 166]}, 'GLZN': {'d': 2, 'p': 65, 'gens': [3844764, 5767146, 7528158, 274951, 7414902, 387908, 3295512]}, 'Perm': {'d': 24, 'gens': [3669247, 3669137, 8023702, 27029669736328361625600, 3669127, 7, 840]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 52], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times S_3\\times C_{52}', 'transitive_degree': 624, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [10, 9, 7], [10, 3, 15], [1, 2, 5], [1, 2, 14], [9, 2, 14], [1, 10, 13], [1, 2, 12]], 'aut_group': '192.1493', 'aut_hash': 1493, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 8, 'aut_perms': [17764, 31287, 5329, 40319, 37965, 21769, 12316], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [4, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '192.1493', 'autcent_hash': 1493, 'autcent_nilpotent': False, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3: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, 8]], 'center_label': '16.10', 'center_order': 16, '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'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 7, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 10, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 16]], 'label': '16.10', 'linC_count': 224, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 48, 'linQ_dim': 4, 'linQ_dim_count': 48, 'linR_count': 48, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C4', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 16, 'number_divisions': 12, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 27, 'number_subgroups': 27, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[10, 9, 7], [10, 3, 15], [1, 2, 5], [1, 2, 14], [9, 2, 14], [1, 10, 13], [1, 2, 12]], 'outer_group': '192.1493', 'outer_hash': 1493, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 8, 'outer_perms': [17764, 31287, 5329, 40319, 37965, 21769, 12316], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4]], 'representations': {'PC': {'code': 516, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7115326, 7115160]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [1173753, 1832002, 438254, 1565004]}, 'Perm': {'d': 8, 'gens': [22, 5040, 120, 7]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
