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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.54046', 'ambient_counter': 54046, 'ambient_order': 256, 'ambient_tex': 'D_4^2:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': True, 'core_order': 64, 'counter': 54, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '256.54046.4.bh1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.bh1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '64.108', 'subgroup_hash': 108, 'subgroup_order': 64, 'subgroup_tex': 'C_2^3.D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.54046', 'aut_centralizer_order': None, 'aut_label': '4.bh1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '32.d1', 'complements': ['64.bg1', '64.bh1', '64.bi1'], 'conjugacy_class_count': 1, 'contained_in': ['2.g1', '2.j1', '2.m1'], 'contains': ['8.k1', '8.l1', '8.n1', '8.cq1', '8.cr1'], 'core': '4.bh1', 'coset_action_label': None, 'count': 1, 'diagramx': [5456, 4829, 5448, 4825], 'generators': [9, 4, 32], 'label': '256.54046.4.bh1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.bh1', 'normal_contained_in': ['2.g1', '2.j1', '2.m1'], 'normal_contains': ['8.k1', '8.l1', '8.n1', '8.cq1', '8.cr1'], 'normalizer': '1.a1', 'old_label': '4.bh1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.bh1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 8, 'aut_gen_orders': [4, 4, 8, 8, 2], 'aut_gens': [[1, 4, 32], [47, 14, 48], [33, 14, 34], [53, 20, 34], [21, 22, 32], [39, 30, 34]], 'aut_group': '1024.dic', 'aut_hash': 6313253630026183879, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 1024, 'aut_permdeg': 20, 'aut_perms': [1776562836247953360, 327412324151182938, 796251669346522927, 151280097934428725, 789850575253328543], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [4, 1, 4, 1], [4, 2, 1, 2], [4, 4, 8, 1], [8, 2, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4.D_4^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [4, 1, 4], [4, 2, 2], [4, 4, 8], [8, 2, 8]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '8.3', 'commutator_count': 1, 'commutator_label': '4.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 108, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [4, 1, 2, 2], [4, 2, 1, 2], [4, 4, 2, 4], [8, 2, 2, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 84, 'exponent': 8, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '8.5', 'hash': 108, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2], 'inner_gens': [[1, 12, 48], [25, 4, 32], [17, 4, 32]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12]], 'label': '64.108', 'linC_count': 64, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 12, 'linQ_dim': 10, 'linQ_dim_count': 8, 'linR_count': 16, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.D4', 'ngens': 3, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 28, 'number_divisions': 18, 'number_normal_subgroups': 41, 'number_subgroup_autclasses': 31, 'number_subgroup_classes': 57, 'number_subgroups': 81, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 40], [8, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 2, 4], 'outer_gen_pows': [0, 0, 32, 0, 32], 'outer_gens': [[1, 20, 50], [1, 6, 50], [37, 22, 32], [1, 4, 48], [5, 22, 34]], 'outer_group': '128.1135', 'outer_hash': 1135, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [125199503, 94525327, 298746857, 136095240, 250835880], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [8, 2]], 'representations': {'PC': {'code': 2288092572549837, 'gens': [1, 3, 6], 'pres': [6, 2, 2, 2, 2, 2, 2, 12, 218, 50, 579, 69, 1733]}, 'GLFq': {'d': 3, 'q': 9, 'gens': [341995342, 137858243, 374880322, 139331306, 86106566, 206422439]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [28679, 31911, 36865, 4225, 13053, 39113]}, 'Perm': {'d': 20, 'gens': [124251769238178969, 276525382685616856, 406278934966084440, 528199954665556440, 16, 656686659640798800]}}, 'schur_multiplier': [2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3.D_4', 'transitive_degree': 32, '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': '32.51', '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, 4, 4, 2, 4, 2], 'aut_gens': [[1, 2, 4, 8, 32], [197, 2, 148, 220, 112], [81, 146, 20, 24, 176], [133, 130, 20, 220, 160], [197, 130, 20, 204, 240], [197, 2, 148, 204, 112], [65, 6, 20, 12, 240], [197, 130, 132, 204, 160], [65, 146, 4, 152, 176], [21, 146, 4, 220, 224]], 'aut_group': None, 'aut_hash': 1847371242188974407, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 16384, 'aut_permdeg': 32, 'aut_perms': [164883162344671318395547255048963767, 36133845032164243504124515961973513, 172353243354807733285757230250005864, 172226414868442194937685868688999858, 164745968027555996974318411271099631, 107982161349786222764682412389266806, 7756089360467022501227610252537752, 167923496825919504719441895008172193, 167978078912057550088421759319465175], 'aut_phi_ratio': 128.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 4, 2, 2], [2, 4, 4, 2], [2, 8, 1, 2], [2, 8, 2, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 2, 2], [4, 4, 4, 1], [4, 8, 1, 2], [4, 8, 2, 3], [8, 4, 2, 2], [8, 8, 2, 3]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '1024.djt', 'autcent_hash': 1957374279625110839, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{10}', '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], [2, 2, 2], [2, 4, 12], [2, 8, 4], [4, 2, 4], [4, 4, 8], [4, 8, 8], [8, 4, 4], [8, 8, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '64.261', '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', '2.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 54046, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 12], [2, 8, 1, 4], [4, 2, 1, 4], [4, 4, 1, 8], [4, 8, 1, 8], [8, 4, 1, 4], [8, 8, 1, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 19998720, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '32.51', 'hash': 54046, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 2, 4], 'inner_gens': [[1, 2, 4, 8, 224], [1, 2, 4, 24, 160], [1, 2, 4, 136, 32], [1, 18, 132, 8, 32], [65, 130, 4, 8, 32]], 'inner_hash': 261, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': False, 'inner_tex': 'D_4\\times C_2^3', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 16], [4, 2], [8, 2]], 'label': '256.54046', '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': 'D4^2:C2^2', 'ngens': 5, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 27, 'number_characteristic_subgroups': 89, 'number_conjugacy_classes': 52, 'number_divisions': 52, 'number_normal_subgroups': 487, 'number_subgroup_autclasses': 534, 'number_subgroup_classes': 1445, 'number_subgroups': 3575, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 87], [4, 104], [8, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4, 4], 'outer_gen_pows': [4, 0, 0, 0, 0, 0], 'outer_gens': [[21, 2, 4, 204, 160], [17, 130, 4, 8, 176], [17, 2, 132, 8, 176], [1, 2, 4, 8, 160], [149, 2, 148, 92, 176], [145, 6, 132, 12, 48]], 'outer_group': '256.29080', 'outer_hash': 29080, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 12, 'outer_perms': [94434623, 175351800, 14923560, 138989520, 182610849, 44770440], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2:D_4^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 20, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 16], [4, 2], [8, 2]], 'representations': {'PC': {'code': 2485853865888185753955113064, 'gens': [1, 2, 3, 4, 6], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 395, 1107, 91, 10757, 3853, 141, 10758, 166]}, 'Perm': {'d': 20, 'gens': [28270632041094965, 123889605643092240, 32389960981200487, 38940201567808560, 288261779527169280, 393462498441893880, 16, 521249814352325640]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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': 'D_4^2:C_2^2', 'transitive_degree': 32, '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': '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, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
