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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1216.13', 'ambient_counter': 13, 'ambient_order': 1216, 'ambient_tex': 'C_{76}.D_8', 'central': True, 'central_factor': False, 'centralizer_order': 1216, 'characteristic': True, 'core_order': 2, 'counter': 63, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1216.13.608.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '608.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '608.50', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 50, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 608, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{76}.D_4', 'simple': True, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': True, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1216.13', 'aut_centralizer_order': 4608, 'aut_label': '608.c1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 4608, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['32.c1.a1', '304.a1.a1', '304.g1.a1'], 'contains': ['1216.a1.a1'], 'core': '608.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5036, 6707, 7720, 2832, 5883, 5133, 6593, 4887], 'generators': [616], 'label': '1216.13.608.c1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '608.c1.a1', 'normal_contained_in': ['32.c1.a1', '304.a1.a1'], 'normal_contains': ['1216.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '608.c1.a1', 'projective_image': '608.50', 'quotient_action_image': '1.1', 'quotient_action_kernel': '608.50', 'quotient_action_kernel_order': 608, 'quotient_fusion': None, 'short_label': '608.c1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', '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]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '152.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 36, 'aut_gen_orders': [2, 4, 2, 2, 2, 4, 18], 'aut_gens': [[1, 2, 16], [1, 6, 16], [5, 2, 16], [1, 618, 16], [609, 2, 16], [1, 918, 624], [317, 2, 624], [1, 2, 336]], 'aut_group': None, 'aut_hash': 2646473537625184211, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 4608, 'aut_permdeg': 50, 'aut_perms': [17144366159230309214391345727105172054222151806104804355796789367, 12277990788770761976510279074755790083060864570638049928487516293, 273552102593913796130398531324241906497813721317740251745972879, 27311179888865291247709923606793574821092642790225856771995075201, 17231278404132749260763623281919532017986397222304707240987950997, 23711346171408062357692861153365870489778299913934405350040128383, 2333242187556313151156684501605716495131430718979431970468552], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 4], [4, 4, 1, 1], [4, 8, 2, 1], [8, 4, 4, 2], [19, 1, 18, 1], [38, 1, 18, 3], [76, 2, 18, 4], [76, 4, 18, 1], [76, 8, 36, 1], [152, 4, 72, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_{18}\\times D_4^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 18, 'autcent_group': '288.840', 'autcent_hash': 840, 'autcent_nilpotent': True, 'autcent_order': 288, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\times C_{18}', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '16.14', 'autcentquo_hash': 14, 'autcentquo_nilpotent': True, 'autcentquo_order': 16, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 4], [4, 4, 1], [4, 8, 2], [8, 4, 8], [19, 1, 18], [38, 1, 54], [76, 2, 72], [76, 4, 18], [76, 8, 36], [152, 4, 144]], 'center_label': '76.4', 'center_order': 76, 'central_product': True, 'central_quotient': '16.3', 'commutator_count': 1, 'commutator_label': '8.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '19.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['19.1', 1], ['64.13', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 4], [4, 4, 1, 1], [4, 8, 1, 2], [8, 4, 4, 2], [19, 1, 18, 1], [38, 1, 18, 3], [76, 2, 18, 4], [76, 4, 18, 1], [76, 8, 18, 2], [152, 4, 72, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 120, 'exponent': 152, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 19], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.2', 'frattini_quotient': '76.4', 'hash': 13, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2], 'inner_gens': [[1, 918, 624], [317, 2, 624], [609, 610, 16]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'C_2^2:C_4', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 152], [2, 190], [4, 19]], 'label': '1216.13', 'linC_count': 5760, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 26, 'linQ_degree_count': 32, 'linQ_dim': 26, 'linQ_dim_count': 8, 'linR_count': 72, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C76.D8', 'ngens': 7, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 24, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 361, 'number_divisions': 26, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 58, 'number_subgroup_classes': 64, 'number_subgroups': 114, 'old_label': None, 'order': 1216, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 28], [8, 32], [19, 18], [38, 54], [76, 504], [152, 576]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 18, 'outer_gen_orders': [2, 2, 2, 2, 18], 'outer_gen_pows': [0, 4, 0, 0, 0], 'outer_gens': [[1, 6, 16], [5, 2, 16], [1, 618, 16], [609, 2, 16], [1, 2, 336]], 'outer_group': '288.840', 'outer_hash': 840, 'outer_nilpotent': True, 'outer_order': 288, 'outer_permdeg': 19, 'outer_perms': [6402373705728000, 20922789888000, 87178291200, 479001600, 3675033], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_{18}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 35, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 19], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5], [18, 4], [36, 4], [72, 5]], 'representations': {'PC': {'code': 11007833477505783180385308968534613141610385, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -19, 4256, 12853, 36, 254, 58, 21844, 10931, 102, 124]}, 'Perm': {'d': 35, 'gens': [27118939565422873430452308456898560000, 313940022231124581682879782256803840000, 617067029315571937081970465062748160000, 854463209869868154716160000, 921484986700374179218773387509760000000, 921484986701649204058770279898644480000, 115242726703104000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 76], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{76}.D_8', 'transitive_degree': 1216, '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': '152.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 72, 'aut_gen_orders': [36, 36, 18, 36, 18], 'aut_gens': [[1, 2, 4], [3, 2, 63], [153, 306, 551], [305, 2, 422], [305, 306, 447], [459, 306, 468]], 'aut_group': None, 'aut_hash': 2339413778631888697, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 2304, 'aut_permdeg': 34, 'aut_perms': [67887505189301186376772055737415367233, 48254851505911670233947116476841944245, 123291195780425269550032038127286995135, 250348622755133829083722180015727578895, 179758167883468341509151625180015940577], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 2, 2, 1], [4, 4, 2, 1], [8, 4, 4, 1], [19, 1, 18, 1], [38, 1, 18, 1], [38, 2, 18, 1], [76, 2, 36, 1], [76, 4, 36, 1], [152, 4, 72, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{18}\\times D_4^2.C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 18, 'autcent_group': '72.18', 'autcent_hash': 18, 'autcent_nilpotent': True, 'autcent_order': 72, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{18}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '32.27', 'autcentquo_hash': 27, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [4, 2, 2], [4, 4, 2], [8, 4, 4], [19, 1, 18], [38, 1, 18], [38, 2, 18], [76, 2, 36], [76, 4, 36], [152, 4, 72]], 'center_label': '38.2', 'center_order': 38, 'central_product': True, 'central_quotient': '16.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '19.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 50, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['19.1', 1], ['32.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 2], [8, 4, 2, 2], [19, 1, 18, 1], [38, 1, 18, 1], [38, 2, 18, 1], [76, 2, 18, 2], [76, 4, 18, 2], [152, 4, 36, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 60, 'exponent': 152, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 19], 'faithful_reps': [[4, 0, 18]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '76.4', 'hash': 50, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 4], 'inner_gens': [[1, 2, 6], [1, 2, 308], [307, 306, 4]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'C_2^2:C_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 72, 'irrQ_dim': 72, 'irrR_degree': 8, 'irrep_stats': [[1, 152], [2, 38], [4, 19]], 'label': '608.50', 'linC_count': 18, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 4, 'linQ_dim': 26, 'linQ_dim_count': 4, 'linR_count': 9, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C76.D4', 'ngens': 6, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 209, 'number_divisions': 18, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 38, 'number_subgroups': 52, 'old_label': None, 'order': 608, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 12], [8, 16], [19, 18], [38, 54], [76, 216], [152, 288]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 36, 'outer_gen_orders': [2, 2, 36], 'outer_gen_pows': [0, 8, 2], 'outer_gens': [[1, 306, 300], [153, 2, 4], [153, 2, 559]], 'outer_group': '144.48', 'outer_hash': 48, 'outer_nilpotent': True, 'outer_order': 144, 'outer_permdeg': 15, 'outer_perms': [3628800, 479001600, 273992861520], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_{18}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 35, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 19], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1], [18, 4], [36, 4], [72, 1]], 'representations': {'PC': {'code': 50528418381872193309528353719121, 'gens': [1, 2, 3], 'pres': [6, -2, -2, 2, -2, -2, -19, 1824, 110, 2780, 50, 7491, 69, 88]}, 'Perm': {'d': 35, 'gens': [3336567538805791502319243669012480000000, 1512245811214635137358154791993999360000, 115242726703104000, 403913035961968660807680000, 903065580851809051163209980724346880000, 295496195434689904216520293126471680000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 76], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{76}.D_4', 'transitive_degree': 304, 'wreath_data': None, 'wreath_product': False}
