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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '864.4704', 'ambient_counter': 4704, 'ambient_order': 864, 'ambient_tex': 'S_3\\times D_6^2', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 108, 'counter': 22, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '864.4704.8.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.b1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '8.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '108.43', 'subgroup_hash': 43, 'subgroup_order': 108, 'subgroup_tex': 'C_3^2:D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '864.4704', 'aut_centralizer_order': 96, 'aut_label': '8.b1', 'aut_quo_index': 42, 'aut_stab_index': 18, 'aut_weyl_group': '288.1031', 'aut_weyl_index': 1728, 'centralizer': '72.b1', 'complements': ['108.b1', '108.a1', '108.g1'], 'conjugacy_class_count': 18, 'contained_in': ['4.a1', '4.b1', '4.c1', '4.e1'], 'contains': ['16.a1', '16.c1', '24.b1', '24.g1', '24.h1'], 'core': '8.b1', 'coset_action_label': None, 'count': 18, 'diagramx': [4535, 4972, 6759, 4191], 'generators': [27461, 40966, 783029, 373141, 27601], 'label': '864.4704.8.b1', 'mobius_quo': 0, 'mobius_sub': -8, 'normal_closure': '8.b1', 'normal_contained_in': ['4.a1', '4.b1', '4.c1', '4.e1'], 'normal_contains': ['16.a1', '16.c1', '24.b1'], 'normalizer': '1.a1', 'old_label': '8.b1', 'projective_image': '432.759', 'quotient_action_image': '4.2', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '8.b1', 'subgroup_fusion': None, 'weyl_group': '72.46'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [4, 3, 6, 3, 2, 6, 2, 4, 2], 'aut_gens': [[1, 6, 18], [1, 84, 24], [43, 6, 18], [11, 6, 90], [73, 6, 18], [55, 6, 18], [91, 6, 30], [5, 12, 90], [49, 78, 96], [13, 12, 90]], 'aut_group': '1728.47845', 'aut_hash': 47845, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1728, 'aut_permdeg': 13, 'aut_perms': [248332440, 2538079200, 621992880, 3154677120, 16, 1140992056, 136095247, 1437182640, 1975534560], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [3, 1, 2, 1], [3, 2, 4, 1], [3, 2, 8, 1], [6, 1, 2, 1], [6, 2, 4, 1], [6, 2, 8, 1], [6, 9, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2:\\GL(2,3)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 1, 2], [3, 2, 12], [6, 1, 2], [6, 2, 12], [6, 9, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 43, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 1, 2, 1], [3, 2, 1, 4], [3, 2, 2, 4], [6, 1, 2, 1], [6, 2, 1, 4], [6, 2, 2, 4], [6, 9, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 273, 'exponent': 6, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '108.43', 'hash': 43, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[1, 12, 90], [13, 6, 18], [37, 6, 18]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 12], [2, 24]], 'label': '108.43', 'linC_count': 144, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 120, 'linQ_dim': 6, 'linQ_dim_count': 120, 'linR_count': 120, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:D6', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 36, 'number_divisions': 24, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 76, 'number_subgroups': 188, 'old_label': None, 'order': 108, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 19], [3, 26], [6, 62]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2], 'outer_gen_pows': [0, 0, 0, 1, 1], 'outer_gens': [[5, 6, 18], [59, 72, 60], [55, 72, 102], [1, 48, 30], [1, 72, 66]], 'outer_group': '96.226', 'outer_hash': 226, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [7, 120, 1456, 11520, 5160], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 12], [4, 8]], 'representations': {'PC': {'code': 72686465195309, 'gens': [1, 3, 4], 'pres': [5, -2, -3, -3, -2, -3, 10, 182, 1803, 58, 1804]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101732552038707, 108470314926331621, 125101729074818783]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [10990224, 8405204, 22274384, 28816400, 29360962]}, 'Perm': {'d': 11, 'gens': [3719526, 1, 8427600, 11980800, 11980830]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:D_6', 'transitive_degree': 36, '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [6, 36, 12], 'aut_gens': [[27611, 27379, 805157, 27751, 715961], [297301, 534601, 805157, 513769, 310976], [32419, 391964, 715961, 534857, 783629], [297001, 40501, 796964, 373447, 535147]], 'aut_group': '497664.i', 'aut_hash': 7795797884732631937, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 497664, 'aut_permdeg': 36, 'aut_perms': [246787933640103821372925485412432292053217, 355294225826174445294504429932927929079677, 362728638386799870223687794249874829082776], 'aut_phi_ratio': 1728.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 12, 1], [2, 9, 12, 1], [2, 27, 4, 1], [3, 2, 3, 1], [3, 4, 3, 1], [3, 8, 1, 1], [6, 2, 9, 1], [6, 4, 9, 1], [6, 6, 24, 1], [6, 8, 3, 1], [6, 12, 12, 1], [6, 18, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^3:C_2^2.S_4^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '384.20164', 'autcent_hash': 20164, 'autcent_nilpotent': False, 'autcent_order': 384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4:S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': '1296.3490', 'autcentquo_hash': 3490, 'autcentquo_nilpotent': False, 'autcentquo_order': 1296, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\wr S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 12], [2, 9, 12], [2, 27, 4], [3, 2, 3], [3, 4, 3], [3, 8, 1], [6, 2, 9], [6, 4, 9], [6, 6, 24], [6, 8, 3], [6, 12, 12], [6, 18, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '216.162', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4704, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['6.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 12], [2, 9, 1, 12], [2, 27, 1, 4], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [6, 2, 1, 9], [6, 4, 1, 9], [6, 6, 1, 24], [6, 8, 1, 3], [6, 12, 1, 12], [6, 18, 1, 12]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 277760000, 'exponent': 6, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '864.4704', 'hash': 4704, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 3, 6, 3], 'inner_gens': [[27611, 27379, 805157, 27151, 715961], [27611, 27379, 464789, 27751, 715961], [27611, 373147, 805157, 27751, 715961], [27011, 27379, 805157, 27751, 310976], [27611, 27379, 805157, 40801, 715961]], 'inner_hash': 162, 'inner_nilpotent': False, 'inner_order': 216, 'inner_split': True, 'inner_tex': 'S_3^3', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 48], [4, 24], [8, 4]], 'label': '864.4704', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'S3*D6^2', '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, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 108, 'number_divisions': 108, 'number_normal_subgroups': 628, 'number_subgroup_autclasses': 162, 'number_subgroup_classes': 3426, 'number_subgroups': 22400, 'old_label': None, 'order': 864, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 255], [3, 26], [6, 582]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 3, 2], 'outer_gen_pows': [27001, 27001, 27089, 27001, 27001, 27001], 'outer_gens': [[27611, 27379, 805157, 783179, 310976], [27611, 27379, 805157, 513769, 715961], [27611, 513541, 643151, 27151, 121969], [783319, 783551, 805157, 783179, 121969], [783551, 783479, 391979, 32483, 297611], [783551, 783319, 783629, 33223, 715961]], 'outer_group': '2304.ez', 'outer_hash': 2138098037908021894, 'outer_nilpotent': False, 'outer_order': 2304, 'outer_permdeg': 12, 'outer_perms': [374578710, 189504870, 11480107, 365150077, 250528384, 1583399], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:S_3^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 48], [4, 24], [8, 4]], 'representations': {'PC': {'code': 2737945798467452556616810884464661525, 'gens': [1, 2, 3, 5, 7], 'pres': [8, -2, -2, -2, -3, -2, -3, -2, -3, 250, 66, 267, 4804, 116, 4613, 1718, 166, 1575]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729074815868, 24992906222183252, 125101746508740793, 91793129339160256, 108378800778161462]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [27451, 27379, 386866, 40966, 297011, 513019, 32773, 27301]}, 'Perm': {'d': 13, 'gens': [40279680, 362887, 721, 1, 362880, 5760, 30, 518918400]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times D_6^2', 'transitive_degree': 48, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 84, 'aut_gen_orders': [3, 3], 'aut_gens': [[1, 2, 4], [4, 5, 3], [2, 4, 1]], 'aut_group': '168.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 7, 'aut_perms': [4361, 244], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 168, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\PSL(2,7)', '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]], 'center_label': '8.5', 'center_order': 8, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8.5', 'hash': 5, '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, 8]], 'label': '8.5', 'linC_count': 28, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 28, 'linQ_dim': 3, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3', 'ngens': 3, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 8, 'number_divisions': 8, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 5, 3], [2, 4, 1]], 'outer_group': '168.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 168, 'outer_permdeg': 7, 'outer_perms': [4361, 244], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -2, 2, 2]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16482, 16322, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 13286, 13933]}, 'Perm': {'d': 6, 'gens': [120, 6, 1]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
