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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '128.1818', 'ambient_counter': 1818, 'ambient_order': 128, 'ambient_tex': 'C_2^4.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 8, 'counter': 40, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '128.1818.8.q1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.q1', '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': '16.10', 'subgroup_hash': 10, 'subgroup_order': 16, 'subgroup_tex': 'C_2^2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '128.1818', 'aut_centralizer_order': None, 'aut_label': '8.q1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '4.k1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['4.f1', '4.k1', '4.o1'], 'contains': ['16.d1', '16.n1', '16.v1'], 'core': '16.d1', 'coset_action_label': None, 'count': 8, 'diagramx': [756, -1, 758, -1], 'generators': [30735, 45581, 37001], 'label': '128.1818.8.q1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.f1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1', 'old_label': '8.q1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.q1', 'subgroup_fusion': None, 'weyl_group': None}
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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}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 4, 4, 4, 2, 4, 4, 4], 'aut_gens': [[30735, 61583, 45581, 61973], [6281, 28807, 62025, 61973], [6153, 28807, 12805, 29341], [63495, 28807, 54915, 64021], [63623, 28679, 7751, 7931], [30735, 61455, 45581, 62101], [30863, 61455, 24195, 24191], [6281, 28679, 64065, 31389], [63623, 61455, 64065, 14873]], 'aut_group': None, 'aut_hash': 8075421528038246761, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 16384, 'aut_permdeg': 48, 'aut_perms': [8450166925671644266600507947205303657911892835319335885906719, 11902319112169768228090937406483398275630339073728435161371189, 11371207724837860607872185399097720925052287719989506262719107, 1587915822310866834537337776832783799157967968387379465523321, 243186844060954130110927397312321409647660625262810958052060, 10282279976434854861544339233202892860786371933011177393901829, 1993784126024293004369821157350219195883401759914073994258562, 5734735071484808203247457112879712133314759413652882971952044], 'aut_phi_ratio': 256.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 4, 1], [2, 2, 4, 1], [4, 2, 2, 2], [4, 4, 2, 1], [4, 4, 8, 1], [4, 8, 2, 2], [8, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 8092891664598236742, 'autcent_nilpotent': True, 'autcent_order': 2048, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8.C_2^3', 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': 8, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 4], [4, 2, 4], [4, 4, 10], [4, 8, 4], [8, 4, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '16.11', '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', '2.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1818, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['64.164', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 4], [4, 2, 1, 4], [4, 4, 1, 2], [4, 4, 2, 4], [4, 8, 1, 4], [8, 4, 2, 4]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 5040, 'exponent': 8, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '16.14', 'hash': 1818, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 1, 2, 4], 'inner_gens': [[30735, 61583, 45581, 29341], [30735, 61583, 45581, 61973], [30735, 61583, 45581, 24191], [63495, 61583, 7879, 61973]], 'inner_hash': 11, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': None, 'inner_tex': 'C_2\\times D_4', 'inner_used': [1, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 20], [4, 2]], 'label': '128.1818', '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': 'C2^4.D4', 'ngens': 4, 'nilpotency_class': 3, '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': 13, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 38, 'number_divisions': 30, 'number_normal_subgroups': 108, 'number_subgroup_autclasses': 88, 'number_subgroup_classes': 230, 'number_subgroups': 412, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 15], [4, 80], [8, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 4, 4, 4, 4, 4], 'outer_gen_pows': [4097, 4097, 4097, 4097, 4097, 4097], 'outer_gens': [[30735, 61455, 47621, 38643], [39041, 61583, 22027, 38643], [6281, 28679, 22027, 64021], [6153, 61455, 54787, 64149], [6153, 61455, 45581, 61973], [63495, 61583, 54787, 38643]], 'outer_group': None, 'outer_hash': 2772261962546327505, 'outer_nilpotent': True, 'outer_order': 1024, 'outer_permdeg': 64, 'outer_perms': [101766017405269312310351413715633019718388648621537060823608700923258499733489997031865412, 36449897692648464440107196777460150234964705132581871229021131256639771109754908839538175, 120938585129702370206620587745113564342922003434159869165437344956279115060852599231748022, 123566215641511313806476181692247121071810643212192631148847572664776359203120139071457860, 45730275922305426883507204193634860263281241066901178132754498882260607761108209408366093, 70810609375535250352815265147184405851497479839373768321419552300820809247570177383254346], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^7.C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4], [4, 10]], 'representations': {'PC': {'code': 4685997646736523828, 'gens': [1, 2, 3, 5], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 58, 844, 438, 102, 1027, 124]}, 'GLZN': {'d': 2, 'p': 24, 'gens': [14113, 159287, 179725, 69125, 47481, 70307, 50883]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [12483, 12953, 4225, 14861, 36873, 39041, 30863]}, 'Perm': {'d': 18, 'gens': [379400881673935, 774492937305989, 85366268440174, 774492937303680, 17764, 1178408162131200, 774492936940800]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4.D_4', 'transitive_degree': 64, 'wreath_data': None, 'wreath_product': False}
