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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '77760.bo', 'ambient_counter': 41, 'ambient_order': 77760, 'ambient_tex': 'C_3^2:D_6\\times S_6', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 3, 'counter': 2570, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '77760.bo.6480.bs2', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6480.bs2', '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': 6480, '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': '12.4', 'subgroup_hash': 4, 'subgroup_order': 12, 'subgroup_tex': 'D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '77760.bo', 'aut_centralizer_order': None, 'aut_label': '6480.bs2', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '810.bz1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['2160.cu2', '2160.cx2', '3240.u2', '3240.v2', '3240.bp2', '3240.ct1', '3240.ct2', '3240.cw1', '3240.cw2'], 'contains': ['12960.k1', '12960.z1', '12960.bn1', '19440.j2'], 'core': '25920.a1', 'coset_action_label': None, 'count': 540, 'diagramx': None, 'generators': [448908007204, 442641301050, 80787], 'label': '77760.bo.6480.bs2', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '135.c1', 'old_label': '6480.bs2', 'projective_image': '77760.bo', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6480.bs2', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [1, 10], [3, 2]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', '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': 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, 1], [2, 3, 2], [3, 2, 1], [6, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 10], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.4', 'linC_count': 1, 'linC_degree': 2, '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': 'D6', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 16, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [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': [[7, 2]], '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': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 43377, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 61, 16, 74]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [17, 35]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 55, 56]}, 'Perm': {'d': 5, 'gens': [6, 1, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_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': 4, 'aut_exponent': 120, 'aut_gen_orders': [12, 12, 30, 30, 12], 'aut_gens': [[138389, 316920, 3634559, 243, 85805, 93924236159, 80880], [357834, 224515, 40002955, 243, 5309, 374183798155, 45363], [321841, 243, 3714605, 143975, 322558, 275469551405, 281788], [5611, 143975, 40055042, 45504, 5161, 449307267842, 148], [138389, 224515, 186814610397, 80644, 5309, 174919403997, 80880], [357362, 45504, 448908047405, 143975, 40892, 274072463405, 281788]], 'aut_group': None, 'aut_hash': 4510848913879562672, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 311040, 'aut_permdeg': 48, 'aut_perms': [2588095353240936216573826954757746300612836352373993210025010, 1671304204921840573303920166558378664303977009252648507248776, 10141641032926069761490728046281796800451983427719401426339696, 1035065277647076057498170696375157234162477637697146392845283, 5205536456841339863165508421544725855479100318961870149291880], 'aut_phi_ratio': 15.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 9, 2, 1], [2, 15, 2, 1], [2, 45, 1, 1], [2, 135, 2, 1], [2, 135, 4, 1], [2, 405, 1, 1], [2, 405, 2, 1], [3, 2, 1, 1], [3, 6, 2, 1], [3, 12, 1, 1], [3, 40, 2, 1], [3, 80, 2, 1], [3, 240, 4, 1], [3, 480, 2, 1], [4, 90, 1, 2], [4, 810, 1, 2], [4, 810, 2, 2], [5, 144, 1, 1], [6, 18, 1, 1], [6, 18, 2, 1], [6, 30, 2, 1], [6, 90, 1, 1], [6, 90, 4, 1], [6, 120, 2, 1], [6, 180, 2, 1], [6, 240, 2, 1], [6, 270, 2, 2], [6, 270, 4, 1], [6, 360, 2, 1], [6, 360, 4, 1], [6, 540, 1, 1], [6, 720, 2, 1], [6, 720, 4, 2], [6, 810, 1, 1], [6, 810, 2, 1], [6, 1080, 2, 1], [6, 1080, 4, 1], [6, 1440, 2, 1], [6, 2160, 2, 1], [6, 2160, 4, 1], [10, 1296, 1, 1], [10, 1296, 2, 1], [12, 180, 1, 2], [12, 540, 2, 2], [12, 1080, 1, 2], [12, 1620, 1, 2], [12, 1620, 2, 2], [15, 288, 1, 1], [15, 864, 2, 1], [15, 1728, 1, 1], [30, 2592, 1, 1], [30, 2592, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3.S_3\\wr C_2.A_6.C_2^2', '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': 120, 'autcentquo_group': None, 'autcentquo_hash': 4510848913879562672, 'autcentquo_nilpotent': False, 'autcentquo_order': 311040, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3.S_3\\wr C_2.A_6.C_2^2', 'cc_stats': [[1, 1, 1], [2, 9, 3], [2, 15, 2], [2, 45, 1], [2, 135, 6], [2, 405, 3], [3, 2, 1], [3, 6, 2], [3, 12, 1], [3, 40, 2], [3, 80, 2], [3, 240, 4], [3, 480, 2], [4, 90, 2], [4, 810, 6], [5, 144, 1], [6, 18, 3], [6, 30, 2], [6, 90, 5], [6, 120, 2], [6, 180, 2], [6, 240, 2], [6, 270, 8], [6, 360, 6], [6, 540, 1], [6, 720, 10], [6, 810, 3], [6, 1080, 6], [6, 1440, 2], [6, 2160, 6], [10, 1296, 3], [12, 180, 2], [12, 540, 4], [12, 1080, 2], [12, 1620, 6], [15, 288, 1], [15, 864, 2], [15, 1728, 1], [30, 2592, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '77760.bo', 'commutator_count': 1, 'commutator_label': '9720.bc', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '360.118'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 41, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['108.17', 1], ['720.763', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 3], [2, 15, 1, 2], [2, 45, 1, 1], [2, 135, 1, 6], [2, 405, 1, 3], [3, 2, 1, 1], [3, 6, 1, 2], [3, 12, 1, 1], [3, 40, 1, 2], [3, 80, 1, 2], [3, 240, 1, 4], [3, 480, 1, 2], [4, 90, 1, 2], [4, 810, 1, 6], [5, 144, 1, 1], [6, 18, 1, 3], [6, 30, 1, 2], [6, 90, 1, 5], [6, 120, 1, 2], [6, 180, 1, 2], [6, 240, 1, 2], [6, 270, 1, 8], [6, 360, 1, 6], [6, 540, 1, 1], [6, 720, 1, 10], [6, 810, 1, 3], [6, 1080, 1, 6], [6, 1440, 1, 2], [6, 2160, 1, 6], [10, 1296, 1, 3], [12, 180, 1, 2], [12, 540, 1, 4], [12, 1080, 1, 2], [12, 1620, 1, 6], [15, 288, 1, 1], [15, 864, 1, 2], [15, 1728, 1, 1], [30, 2592, 1, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 364476672, 'exponent': 60, 'exponents_of_order': [6, 5, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[30, 1, 8], [54, 1, 4], [60, 1, 4], [96, 1, 2]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '25920.w', 'hash': 4343203790838598385, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [6, 3, 2, 3, 2, 10, 3], 'inner_gens': [[138389, 224515, 3986397, 80644, 5309, 93924587997, 80880], [357834, 316920, 3812645, 45504, 40585, 93924414245, 45363], [357834, 179012, 3634559, 243, 40585, 93967781759, 45363], [219122, 362187, 3634559, 243, 85946, 93924236159, 80880], [218885, 281788, 3669692, 148, 85805, 93924271292, 45504], [357834, 179012, 442641029759, 243, 40585, 93924236159, 45363], [138389, 281788, 3669692, 243, 5309, 93924271292, 80880]], 'inner_hash': 4343203790838598385, 'inner_nilpotent': False, 'inner_order': 77760, 'inner_split': True, 'inner_tex': 'C_3^2:D_6\\times S_6', 'inner_used': [1, 2, 3, 5, 6], 'irrC_degree': 30, 'irrQ_degree': 30, 'irrQ_dim': 30, 'irrR_degree': 30, 'irrep_stats': [[1, 8], [2, 8], [4, 2], [5, 16], [6, 4], [9, 8], [10, 24], [16, 4], [18, 8], [20, 12], [30, 8], [32, 4], [36, 2], [40, 2], [54, 4], [60, 4], [64, 1], [96, 2]], 'label': '77760.bo', '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': False, 'metacyclic': False, 'monomial': False, 'name': 'C3^2:D6*S6', '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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 64, 'number_characteristic_subgroups': 17, 'number_conjugacy_classes': 121, 'number_divisions': 121, 'number_normal_subgroups': 41, 'number_subgroup_autclasses': 2849, 'number_subgroup_classes': 7808, 'number_subgroups': 2059265, 'old_label': None, 'order': 77760, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 2127], [3, 2186], [4, 5040], [5, 144], [6, 38454], [10, 3888], [12, 14400], [15, 3744], [30, 7776]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 442680940800], 'outer_gens': [[138718, 80644, 3669385, 362187, 86155, 93924270985, 224515], [322413, 179012, 199348168892, 243, 85805, 193560232892, 45363]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 2], [5, 16], [6, 4], [9, 8], [10, 24], [16, 4], [18, 8], [20, 12], [30, 8], [32, 4], [36, 2], [40, 2], [54, 4], [60, 4], [64, 1], [96, 2]], 'representations': {'Perm': {'d': 15, 'gens': [138389, 316920, 3634559, 243, 85805, 93924236159, 80880]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:D_6\\times S_6', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
