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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '3888.by', 'ambient_counter': 51, 'ambient_order': 3888, 'ambient_tex': 'C_3^3:(S_3\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 81, 'counter': 14, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '3888.by.6.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.f1.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': 6, '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': '648.722', 'subgroup_hash': 722, 'subgroup_order': 648, 'subgroup_tex': 'C_3^4:D_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3888.by', 'aut_centralizer_order': 1, 'aut_label': '6.f1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '2592.fs', 'aut_weyl_index': 3, 'centralizer': '3888.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.b1.a1'], 'contains': ['12.d1.a1', '12.f1.a1', '12.h1.a1', '18.h1.a1', '18.j1.a1', '54.b1.a1'], 'core': '48.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [8567, -1, 8248, -1, 7366, -1, 1770, -1], 'generators': [1139040, 18547920, 10801, 99514800, 19686963, 44312426, 92025360], 'label': '3888.by.6.f1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '6.f1.a1', 'projective_image': '3888.by', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.f1.a1', 'subgroup_fusion': None, 'weyl_group': '1296.3530'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [6, 6, 12], 'aut_gens': [[85816, 40284863, 806400, 7706160, 84228480, 325, 435], [7621139, 131725978, 171319680, 7706160, 84228480, 325, 348], [766739, 40284863, 806400, 84228480, 7706160, 325, 435], [215274213, 83462536, 83548080, 87897600, 170639280, 348, 458]], 'aut_group': '5184.gk', 'aut_hash': 2408390580534296986, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5184, 'aut_permdeg': 36, 'aut_perms': [60355223328790872484488616527198716724990, 217306508066547805258572707799075225542528, 354174434492318837711567880491056347894740], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 18, 2, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 4, 4, 1], [3, 8, 2, 1], [3, 8, 4, 1], [4, 162, 1, 1], [6, 18, 2, 1], [6, 18, 4, 1], [6, 36, 1, 1], [6, 36, 2, 1], [6, 36, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_3^2\\times S_3^2):\\SD_{16}', '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': 24, 'autcentquo_group': '5184.gk', 'autcentquo_hash': 2408390580534296986, 'autcentquo_nilpotent': False, 'autcentquo_order': 5184, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(C_3^2\\times S_3^2):\\SD_{16}', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 18, 2], [3, 2, 2], [3, 4, 7], [3, 8, 6], [4, 162, 1], [6, 18, 6], [6, 36, 7]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '648.722', 'commutator_count': 1, 'commutator_label': '162.52', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 722, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 18, 1, 2], [3, 2, 1, 2], [3, 4, 1, 3], [3, 4, 2, 2], [3, 8, 1, 2], [3, 8, 2, 2], [4, 162, 1, 1], [6, 18, 1, 2], [6, 18, 2, 2], [6, 36, 1, 3], [6, 36, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 12, 'exponent': 12, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '648.722', 'hash': 722, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 2, 3, 3, 3, 3], 'inner_gens': [[85816, 41091263, 806400, 14883120, 84228480, 325, 348], [766216, 40284863, 806400, 84228480, 7706160, 458, 435], [85816, 40284863, 806400, 14883120, 163336320, 325, 435], [14923576, 131725583, 15689520, 7706160, 84228480, 325, 435], [85816, 207290303, 163739520, 7706160, 84228480, 325, 435], [85816, 40285413, 806400, 7706160, 84228480, 325, 435], [86339, 40284863, 806400, 7706160, 84228480, 325, 435]], 'inner_hash': 722, 'inner_nilpotent': False, 'inner_order': 648, 'inner_split': False, 'inner_tex': 'C_3^4:D_4', 'inner_used': [1, 2, 4, 6, 7], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 4], [2, 9], [4, 14], [8, 6]], 'label': '648.722', 'linC_count': 32, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 24, 'linQ_dim': 8, 'linQ_dim_count': 20, 'linR_count': 20, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^4:D4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 15, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 33, 'number_divisions': 25, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 92, 'number_subgroup_classes': 166, 'number_subgroups': 1508, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 45], [3, 80], [4, 162], [6, 360]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[85816, 40284863, 806400, 7706160, 84228480, 325, 348], [40284863, 85816, 806400, 95477760, 87897600, 435, 458]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [4, 8], [8, 6], [16, 2]], 'representations': {'PC': {'code': 225310303518138221091178824050213676311, 'gens': [1, 2, 4, 6, 7], 'pres': [7, -2, -2, -3, -2, -3, -3, 3, 645, 36, 170, 850, 80, 851, 4548, 530, 21174, 1777, 1791]}, 'Perm': {'d': 12, 'gens': [85816, 40284863, 806400, 7706160, 84228480, 325, 435]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:D_4', 'transitive_degree': 24, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [6, 12, 2], 'aut_gens': [[88316188, 44312426], [210612484, 283829906], [296293828, 161335465], [40815387, 101790149]], 'aut_group': '7776.be', 'aut_hash': 4734684673362351397, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7776, 'aut_permdeg': 81, 'aut_perms': [4296961770355805178712717212955004737694318774776262567125953076704460943789512972329148323616490402382614961428531606518, 2737790705357944306175170637752959418613150790561247614753023021154747563798774390016992732512123366802266358727547123931, 562643515008135466600752998718950791465451763653925058474313586234372103183257346028266550703481854041621491587557808859], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 54, 1, 2], [2, 81, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 2], [3, 16, 1, 1], [3, 24, 1, 1], [3, 72, 1, 1], [3, 144, 1, 1], [4, 54, 1, 1], [4, 486, 1, 1], [6, 54, 1, 2], [6, 108, 1, 7], [6, 216, 1, 2], [6, 648, 1, 1], [9, 144, 1, 1], [9, 144, 2, 1], [12, 108, 1, 2], [12, 108, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3^4:S_3', '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': 36, 'autcentquo_group': '7776.be', 'autcentquo_hash': 4734684673362351397, 'autcentquo_nilpotent': False, 'autcentquo_order': 7776, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^4:S_3', 'cc_stats': [[1, 1, 1], [2, 27, 2], [2, 54, 2], [2, 81, 1], [3, 2, 1], [3, 6, 1], [3, 8, 1], [3, 12, 2], [3, 16, 1], [3, 24, 1], [3, 72, 1], [3, 144, 1], [4, 54, 1], [4, 486, 1], [6, 54, 2], [6, 108, 7], [6, 216, 2], [6, 648, 1], [9, 144, 3], [12, 108, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '3888.by', 'commutator_count': 1, 'commutator_label': '972.877', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 51, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 54, 1, 2], [2, 81, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 2], [3, 16, 1, 1], [3, 24, 1, 1], [3, 72, 1, 1], [3, 144, 1, 1], [4, 54, 1, 1], [4, 486, 1, 1], [6, 54, 1, 2], [6, 108, 1, 7], [6, 216, 1, 2], [6, 648, 1, 1], [9, 144, 1, 1], [9, 144, 2, 1], [12, 108, 1, 2], [12, 108, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 234, 'exponent': 36, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 2], [12, 1, 2], [16, 0, 2], [16, 1, 1], [24, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '3888.by', 'hash': 1872573129080097874, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [12, 4], 'inner_gens': [[88316188, 335066669], [335076627, 44312426]], 'inner_hash': 1872573129080097874, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': True, 'inner_tex': 'C_3^3:(S_3\\times S_4)', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 6], [8, 2], [12, 9], [16, 4], [24, 2]], 'label': '3888.by', 'linC_count': 8, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3:(S3*S4)', 'ngens': 2, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 34, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 36, 'number_divisions': 34, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 437, 'number_subgroup_classes': 449, 'number_subgroups': 20056, 'old_label': None, 'order': 3888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 243], [3, 296], [4, 540], [6, 1944], [9, 432], [12, 432]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[210567124, 55607066]], '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': 6, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 6], [8, 2], [12, 7], [16, 2], [24, 3], [32, 1]], 'representations': {'PC': {'code': '1283379239686867955252878143427145393711294631476915545020487158396020491830501428234325265646201755223', 'gens': [1, 2, 4, 6, 8, 9], 'pres': [9, 2, 2, 3, 2, 3, 2, 3, 3, 3, 8316, 44029, 46, 16742, 35436, 29289, 102, 1093, 68045, 65138, 45221, 9428, 158, 217734, 54447, 6828, 1545, 10384, 62233, 916, 81656, 23345, 40850, 20447, 2969]}, 'Perm': {'d': 12, 'gens': [88316188, 44312426]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:(S_3\\times S_4)', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
