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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1248.1282', 'ambient_counter': 1282, 'ambient_order': 1248, 'ambient_tex': 'C_{78}:C_4^2', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 26, 'counter': 49, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1248.1282.48.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '48.c1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '48.20', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 20, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 48, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4\\times C_{12}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '26.1', 'subgroup_hash': 1, 'subgroup_order': 26, 'subgroup_tex': 'D_{13}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1248.1282', 'aut_centralizer_order': 64, 'aut_label': '48.c1', 'aut_quo_index': 3, 'aut_stab_index': 2, 'aut_weyl_group': '156.7', 'aut_weyl_index': 128, 'centralizer': '52.a1', 'complements': ['26.c1'], 'conjugacy_class_count': 2, 'contained_in': ['16.c1', '24.b1', '24.f1'], 'contains': ['96.a1', '624.c1'], 'core': '48.c1', 'coset_action_label': None, 'count': 2, 'diagramx': [5841, 5616, 6346, 4852], 'generators': [824, 96], 'label': '1248.1282.48.c1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '48.c1', 'normal_contained_in': ['16.c1', '24.b1', '24.f1'], 'normal_contains': ['96.a1'], 'normalizer': '1.a1', 'old_label': '48.c1', 'projective_image': '1248.1282', 'quotient_action_image': '4.1', 'quotient_action_kernel': '12.2', 'quotient_action_kernel_order': 12, 'quotient_fusion': None, 'short_label': '48.c1', 'subgroup_fusion': None, 'weyl_group': '52.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 156, 'aut_gen_orders': [12, 13], 'aut_gens': [[1, 2], [1, 14], [3, 2]], 'aut_group': '156.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 156, 'aut_permdeg': 13, 'aut_perms': [4790757360, 570218733], 'aut_phi_ratio': 13.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 13, 1, 1], [13, 2, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_{13}', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 156, 'autcentquo_group': '156.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 156, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{13}', 'cc_stats': [[1, 1, 1], [2, 13, 1], [13, 2, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '26.1', 'commutator_count': 1, 'commutator_label': '13.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '13.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 13, 1, 1], [13, 2, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 26, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 13], 'faithful_reps': [[2, 1, 6]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '26.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 26, 'inner_gen_orders': [2, 13], 'inner_gens': [[1, 24], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 26, 'inner_split': False, 'inner_tex': 'D_{13}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 6]], 'label': '26.1', 'linC_count': 6, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 1, 'linQ_dim': 12, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D13', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 8, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 16, 'old_label': None, 'order': 26, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 13], [13, 12]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [1], 'outer_gens': [[1, 4]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [12, 1]], 'representations': {'PC': {'code': 6635, 'gens': [1, 2], 'pres': [2, -2, -13, 97]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [2211, 2209]}, 'Perm': {'d': 13, 'gens': [479001599, 5748019200]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{13}', 'transitive_degree': 13, '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': 2, 'aut_exponent': 156, 'aut_gen_orders': [6, 12, 12, 12, 12, 12], 'aut_gens': [[1, 4, 16], [1, 868, 882], [3, 630, 944], [625, 6, 592], [1, 967, 112], [625, 1014, 112], [627, 727, 466]], 'aut_group': None, 'aut_hash': 603711047967072108, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 19968, 'aut_permdeg': 106, 'aut_perms': [102635702104063889510876254118212861627754279038412156038602194907309879714850603534638323986109401854142203377981721182711626161153674479104014065023095758769302412722560, 85096284130215922222847885336030709233566906574856427386314353398971218035964971380300877085999019907158912413288210443609056522858022695618680419431634350044659793095905, 5141357231769106974662902284153629216622458944684413883021159409192323235538008493690785677029152652495009096198620174561089836846134259776990330828660644013911636299822, 95021620406095128383088495175889276110613822027447256442945135058597971043175598697129423430221325518301809568025750793872610588940915401572590977024176981345143205670518, 108487999207537710608868703740202778973497091339518781160295668653047440267135569278178227374955195861854958127659359163939244249274332860971901254313367959697254774715541, 84106864159309274748768720223225223418446248151913747568950753942049690245012189799782160675838945857935527065718623257149810806140512240039286544297790964594005342050657], 'aut_phi_ratio': 52.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 13, 2, 2], [3, 1, 2, 1], [4, 1, 4, 1], [4, 13, 4, 1], [4, 13, 8, 2], [6, 1, 2, 1], [6, 1, 4, 1], [6, 13, 4, 2], [12, 1, 8, 1], [12, 13, 8, 1], [12, 13, 16, 2], [13, 4, 3, 1], [26, 4, 3, 1], [26, 4, 6, 1], [39, 4, 6, 1], [52, 4, 12, 1], [78, 4, 6, 1], [78, 4, 12, 1], [156, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{26}.(C_2^3\\times C_6).C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1755', 'autcent_hash': 1755, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4:D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 156, 'autcentquo_group': '156.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 156, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{13}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 13, 4], [3, 1, 2], [4, 1, 4], [4, 13, 20], [6, 1, 6], [6, 13, 8], [12, 1, 8], [12, 13, 40], [13, 4, 3], [26, 4, 9], [39, 4, 6], [52, 4, 12], [78, 4, 18], [156, 4, 24]], 'center_label': '24.9', 'center_order': 24, 'central_product': True, 'central_quotient': '52.3', 'commutator_count': 1, 'commutator_label': '13.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '13.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1282, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['4.1', 1], ['52.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 13, 1, 4], [3, 1, 2, 1], [4, 1, 2, 2], [4, 13, 2, 10], [6, 1, 2, 3], [6, 13, 2, 4], [12, 1, 4, 2], [12, 13, 4, 10], [13, 4, 3, 1], [26, 4, 3, 3], [39, 4, 6, 1], [52, 4, 6, 2], [78, 4, 6, 3], [156, 4, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 30576, 'exponent': 156, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 3, 13], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '624.249', 'hash': 1282, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 52, 'inner_gen_orders': [1, 4, 13], 'inner_gens': [[1, 4, 16], [1, 4, 1168], [1, 100, 16]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 52, 'inner_split': True, 'inner_tex': 'C_{13}:C_4', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 96], [4, 72]], 'label': '1248.1282', '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': 'C78:C4^2', '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': 26, 'number_characteristic_subgroups': 28, 'number_conjugacy_classes': 168, 'number_divisions': 52, 'number_normal_subgroups': 124, 'number_subgroup_autclasses': 92, 'number_subgroup_classes': 216, 'number_subgroups': 1320, 'old_label': None, 'order': 1248, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 55], [3, 2], [4, 264], [6, 110], [12, 528], [13, 12], [26, 36], [39, 24], [52, 48], [78, 72], [156, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 3, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 12, 0, 0, 0], 'outer_gens': [[627, 4, 400], [3, 631, 848], [3, 6, 850], [1, 4, 1232], [1, 4, 112], [1, 630, 16], [3, 4, 400], [1, 6, 400]], 'outer_group': '384.17463', 'outer_hash': 17463, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 13, 'outer_perms': [1607599464, 495099384, 2083656984, 2087286504, 3, 1158363360, 1963543680, 2087286480], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\wr C_2^2\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 12], [12, 4], [24, 6], [48, 2]], 'representations': {'PC': {'code': 388736814627878173738769460894450555284471831, 'gens': [1, 3, 5], 'pres': [7, -2, -2, -2, -2, -2, -3, -13, 14, 58, 10238, 1775, 102, 11443, 4226, 166, 9428, 7083]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [2273949172, 2629817970, 4079222814, 9789111396, 2161971291, 6526074264, 3849240483]}, 'Perm': {'d': 22, 'gens': [2568424348805817727, 127, 137, 45360, 5232169043072294400, 7, 58877387640920217600]}}, '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_{78}:C_4^2', 'transitive_degree': 312, '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': '48.20', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 4, 6, 2, 2], 'aut_gens': [[1, 4], [25, 30], [27, 21], [36, 23], [25, 6], [3, 30]], 'aut_group': '192.1488', 'aut_hash': 1488, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 10, 'aut_perms': [126, 5190, 80641, 806400, 367920], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [4, 1, 12, 1], [6, 1, 6, 1], [12, 1, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times \\GL(2,\\mathbb{Z}/4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '192.1488', 'autcent_hash': 1488, 'autcent_nilpotent': False, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times \\GL(2,\\mathbb{Z}/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, 3], [3, 1, 2], [4, 1, 12], [6, 1, 6], [12, 1, 24]], 'center_label': '48.20', 'center_order': 48, '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', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 20, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [4, 1, 2, 6], [6, 1, 2, 3], [12, 1, 4, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '12.5', 'hash': 20, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 4], [1, 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, 48]], 'label': '48.20', 'linC_count': 384, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 72, 'linQ_dim': 6, 'linQ_dim_count': 72, 'linR_count': 96, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4*C12', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 48, 'number_divisions': 20, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 30, 'number_subgroups': 30, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 12], [6, 6], [12, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 4, 6, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[25, 30], [27, 21], [36, 23], [25, 6], [3, 30]], 'outer_group': '192.1488', 'outer_hash': 1488, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 10, 'outer_perms': [126, 5190, 80641, 806400, 367920], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times \\GL(2,\\mathbb{Z}/4)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 6]], 'representations': {'PC': {'code': 1289748897, 'gens': [1, 3], 'pres': [5, -2, -2, -2, -2, -3, 10, 42, 58]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101739150919715, 91655879435939606]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [10993, 15383]}, 'Perm': {'d': 11, 'gens': [11612160, 2400, 4, 3669120, 744]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 12], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times C_{12}', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
