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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '576.5406', 'ambient_counter': 5406, 'ambient_order': 576, 'ambient_tex': 'C_2^3.\\SOPlus(4,2)', 'central': False, 'central_factor': False, 'centralizer_order': 144, 'characteristic': False, 'core_order': 2, 'counter': 238, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '576.5406.96.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '96.e1.a1', 'outer_equivalence': False, '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': 96, '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': '6.2', 'subgroup_hash': 2, 'subgroup_order': 6, 'subgroup_tex': 'C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '576.5406', 'aut_centralizer_order': 1152, 'aut_label': '96.e1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '2.1', 'aut_weyl_index': 2304, 'centralizer': '4.h1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['32.c1.a1', '48.a1.a1', '48.w1.a1', '48.y1.a1', '48.z1.a1'], 'contains': ['192.a1.a1', '288.c1.a1'], 'core': '288.c1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [912, -1, 5970, -1, 2838, -1, 5984, -1], 'generators': [56, 416], 'label': '576.5406.96.e1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '32.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '96.e1.a1', 'projective_image': '288.887', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '96.e1.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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [5]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], '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]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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]], 'center_label': '6.2', 'center_order': 6, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['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]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 6]], 'label': '6.2', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 6, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2]], 'representations': {'PC': {'code': 21, 'gens': [1], 'pres': [2, -2, -3, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [73]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 56]}, 'Perm': {'d': 5, 'gens': [24, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6', 'transitive_degree': 6, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [4, 4, 6, 6, 6, 6, 6, 6, 6], 'aut_gens': [[1, 2, 16, 96], [77, 458, 400, 544], [61, 242, 272, 128], [61, 270, 272, 96], [13, 270, 400, 536], [225, 230, 16, 504], [401, 454, 16, 496], [93, 198, 400, 528], [53, 430, 272, 152], [29, 238, 272, 144]], 'aut_group': None, 'aut_hash': 8050331733301323351, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4608, 'aut_permdeg': 52, 'aut_perms': [25768534046731414169845977198498801682276923890485726331895631667880, 18804588061362678213284279845721960132438226349592609058203411654876, 60488170151394466897290818993551693158343647742522177666067509884374, 33550030853843807867417486758394403601660449130038674233164596656249, 4405596204934752372501548455564702743915958861264721748736326797203, 28244528234448291175408301256845624443979244773138712387428859509392, 61342497888018224612903521128263400250192940297924774700951640989862, 62007642196034831782814953845341398389808601071252685657542179547211, 61885482316313761164236921350220990706378029626861423728555732496720], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 12, 2, 1], [3, 4, 1, 2], [4, 12, 2, 1], [4, 18, 1, 2], [4, 18, 2, 1], [4, 24, 1, 2], [6, 4, 1, 6], [6, 8, 2, 2], [6, 24, 2, 1], [8, 36, 2, 2], [12, 24, 2, 3]], 'aut_supersolvable': False, 'aut_tex': 'C_3:S_3.C_2^6.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '72.40', 'autcentquo_hash': 40, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 1], [2, 12, 2], [3, 4, 2], [4, 12, 2], [4, 18, 4], [4, 24, 2], [6, 4, 6], [6, 8, 4], [6, 24, 2], [8, 36, 4], [12, 24, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '144.186', 'commutator_count': 2, 'commutator_label': '72.34', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 5406, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 12, 1, 2], [3, 4, 1, 2], [4, 12, 1, 2], [4, 18, 1, 2], [4, 18, 2, 1], [4, 24, 1, 2], [6, 4, 1, 6], [6, 8, 1, 4], [6, 24, 1, 2], [8, 36, 2, 2], [12, 24, 1, 2], [12, 24, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 13440, 'exponent': 24, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '144.186', 'hash': 5406, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 3, 6], 'inner_gens': [[1, 6, 16, 504], [13, 2, 272, 176], [1, 418, 16, 96], [281, 18, 16, 96]], 'inner_hash': 186, 'inner_nilpotent': False, 'inner_order': 144, 'inner_split': True, 'inner_tex': 'S_3^2:C_2^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 10], [4, 17], [8, 4]], 'label': '576.5406', 'linC_count': 48, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.SO+(4,2)', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 28, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 39, 'number_divisions': 34, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 206, 'number_subgroup_classes': 271, 'number_subgroups': 1562, 'old_label': None, 'order': 576, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 31], [3, 8], [4, 144], [6, 104], [8, 144], [12, 144]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 4, 0], 'outer_gens': [[9, 2, 16, 96], [1, 10, 16, 96], [1, 2, 16, 104], [5, 2, 400, 544], [57, 2, 16, 96]], 'outer_group': '32.51', 'outer_hash': 51, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 13], [8, 7]], 'representations': {'PC': {'code': 1295873942959053242059640485982562766427069042187536815877, 'gens': [1, 2, 5, 7], 'pres': [8, 2, 2, 2, 2, 2, 3, 2, 3, 64, 97, 41, 290, 66, 5452, 820, 116, 10765, 789, 28230, 4942, 6742, 166, 26631, 8207, 6167]}, 'Perm': {'d': 26, 'gens': [2299216350016143749640, 15537113421176574734628965, 32390573712065287, 53952440847693490252200, 16, 521647785267532560, 1269047912603090780160000, 31695696522600758968320000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3.\\SOPlus(4,2)', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
