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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1184.197', 'ambient_counter': 197, 'ambient_order': 1184, 'ambient_tex': 'C_{74}.C_4^2', 'central': False, 'central_factor': False, 'centralizer_order': 296, 'characteristic': True, 'core_order': 37, 'counter': 38, 'cyclic': True, 'direct': False, 'hall': 37, 'label': '1184.197.32.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '32.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '32.2', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 32, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2.C_4^2', 'simple': True, 'solvable': True, 'special_labels': ['L2', 'C5'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '37.1', 'subgroup_hash': 1, 'subgroup_order': 37, 'subgroup_tex': 'C_{37}', 'supersolvable': True, 'sylow': 37}
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gps_subgroup_data • Show schema
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{'ambient': '1184.197', 'aut_centralizer_order': 1184, 'aut_label': '32.a1', 'aut_quo_index': 12, 'aut_stab_index': 1, 'aut_weyl_group': '36.2', 'aut_weyl_index': 1184, 'centralizer': '4.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.a1.a1', '16.b1.a1', '16.c1.a1', '16.d1.a1', '16.d1.b1', '16.e1.a1', '16.e1.b1'], 'contains': ['1184.a1.a1'], 'core': '32.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4567, 3772, 3444, 3616, 5689, 4353, 3848, 4396], 'generators': [32], 'label': '1184.197.32.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '32.a1.a1', 'normal_contained_in': ['16.c1.a1', '16.a1.a1', '16.b1.a1', '16.e1.b1', '16.e1.a1', '16.d1.a1', '16.d1.b1'], 'normal_contains': ['1184.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '32.a1.a1', 'projective_image': '1184.197', 'quotient_action_image': '4.1', 'quotient_action_kernel': '8.2', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '32.a1.a1', 'subgroup_fusion': None, 'weyl_group': '4.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '37.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 36, 'aut_gen_orders': [36], 'aut_gens': [[1], [2]], 'aut_group': '36.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 36, 'aut_permdeg': 13, 'aut_perms': [522593433], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [37, 1, 36, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{36}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 36, 'autcent_group': '36.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{36}', '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], [37, 1, 36]], 'center_label': '37.1', 'center_order': 37, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['37.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [37, 1, 36, 1]], 'element_repr_type': 'PC', 'elementary': 37, 'eulerian_function': 1, 'exponent': 37, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [37], 'faithful_reps': [[1, 0, 36]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '37.1', 'hash': 1, 'hyperelementary': 37, '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': 36, 'irrQ_dim': 36, 'irrR_degree': 2, 'irrep_stats': [[1, 37]], 'label': '37.1', 'linC_count': 36, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 36, 'linQ_degree_count': 1, 'linQ_dim': 36, 'linQ_dim_count': 1, 'linR_count': 18, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C37', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 37, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 37, 'order_factorization_type': 1, 'order_stats': [[1, 1], [37, 36]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 36, 'outer_gen_orders': [36], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '36.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 36, 'outer_permdeg': 13, 'outer_perms': [522593433], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{36}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 37, 'pgroup': 37, 'primary_abelian_invariants': [37], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [36, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -37]}, 'Lie': [{'d': 1, 'q': 37, 'gens': [382630662501032184766604355445682020940313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 37, 'gens': [50691]}, 'Perm': {'d': 37, 'gens': [13391759764436443828847980133430067200000000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [37], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{37}', 'transitive_degree': 37, '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': '16.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1332, 'aut_gen_orders': [36, 12, 18, 18, 36], 'aut_gens': [[1, 4, 8], [833, 4, 140], [709, 4, 1004], [853, 4, 568], [513, 4, 1132], [681, 4, 428]], 'aut_group': None, 'aut_hash': 5694167456236791792, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 42624, 'aut_permdeg': 296, 'aut_perms': 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35349477964771770149442140872384695349165003798581712170351793208580977580358677561915406080637383582396632511428911565873691583255434184888334923150488349515383051742408277693559536545079022471975063534246705597027684411300971346026059208517744774704038482403153907726656974611894033581679851559611488333499739143842789029600541738575565454790250039783267647192744630711207238678268318574957451648772899322180593130789860208910613584196200904977726067447178085769212007035796423946925625929142676696938425659280378617334326434383452675131632715242444828618837567538229676680636656596745477574111114661366], 'aut_phi_ratio': 74.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 37, 2, 2], [4, 2, 2, 1], [4, 74, 2, 1], [4, 74, 4, 2], [37, 4, 9, 1], [74, 4, 9, 3], [148, 4, 36, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{74}.C_{18}.C_4.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1332, 'autcentquo_group': '2664.b', 'autcentquo_hash': 7967020058022097167, 'autcentquo_nilpotent': False, 'autcentquo_order': 2664, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_{37}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 37, 4], [4, 2, 2], [4, 74, 10], [37, 4, 9], [74, 4, 27], [148, 4, 36]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '296.12', 'commutator_count': 1, 'commutator_label': '74.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '37.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 197, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 37, 1, 4], [4, 2, 2, 1], [4, 74, 2, 5], [37, 4, 9, 1], [74, 4, 9, 3], [148, 4, 36, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 148, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 37], 'factors_of_order': [2, 37], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '296.12', 'hash': 197, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 148, 'inner_gen_orders': [4, 1, 74], 'inner_gens': [[1, 4, 940], [1, 4, 8], [261, 4, 8]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 296, 'inner_split': False, 'inner_tex': 'C_{74}:C_4', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 4], [4, 72]], 'label': '1184.197', 'linC_count': 288, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 40, 'linQ_degree_count': 40, 'linQ_dim': 40, 'linQ_dim_count': 28, 'linR_count': 90, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C74.C4^2', 'ngens': 6, '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': 15, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 92, 'number_divisions': 19, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 76, 'number_subgroups': 1612, 'old_label': None, 'order': 1184, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 151], [4, 744], [37, 36], [74, 108], [148, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 36, 'outer_gen_orders': [2, 2, 36], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 4, 588], [1, 4, 1176], [297, 4, 520]], 'outer_group': '144.48', 'outer_hash': 48, 'outer_nilpotent': True, 'outer_order': 144, 'outer_permdeg': 15, 'outer_perms': [3628800, 12936672000, 93888034864], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_{18}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 45, 'pgroup': 0, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [36, 2], [72, 3]], 'representations': {'PC': {'code': 308797559696454112260347534640847754661923, 'gens': [1, 3, 4], 'pres': [6, -2, -2, -2, 2, -2, -37, 12, 22563, 7017, 69, 20644, 17530, 88, 6917, 20747]}, 'GLZN': {'d': 2, 'p': 111, 'gens': [1367965, 144065451, 1367704, 99837136, 51970016, 1367699]}, 'Perm': {'d': 45, 'gens': [61889624625147994221916347489785178302976904550346367, 1577, 11520, 125048811375705151205535935738814025741110415178467200, 7, 2845141573787144435826726013057439325431400030470246400]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{74}.C_4^2', 'transitive_degree': 296, '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': '16.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 3, 2, 2, 2, 2], 'aut_gens': [[1, 4, 8], [29, 4, 24], [21, 4, 26], [7, 4, 28], [25, 4, 23], [19, 4, 28], [7, 4, 26], [5, 4, 12], [5, 4, 8]], 'aut_group': '384.20100', 'aut_hash': 20100, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 12, 'aut_perms': [235050896, 424023308, 183902764, 207622653, 140104852, 177005226, 165757252, 182450408], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 3, 2], [4, 2, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\wr S_3', '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': 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, 7], [4, 2, 12]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 2, 2, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 1, 2], 'inner_gens': [[1, 4, 12], [1, 4, 8], [5, 4, 8]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 4]], 'label': '32.2', 'linC_count': 192, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 76, 'linQ_dim': 6, 'linQ_dim_count': 49, 'linR_count': 49, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2.C4^2', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, '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': 20, 'number_divisions': 14, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 13, 'number_subgroup_classes': 38, 'number_subgroups': 50, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 24]], '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, 0, 0], 'outer_gens': [[3, 4, 9], [3, 4, 24], [30, 4, 25], [17, 4, 10], [7, 4, 30]], 'outer_group': '96.195', 'outer_hash': 195, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [127, 23, 1442, 11520, 5160], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,\\mathbb{Z}/4)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10]], 'representations': {'PC': {'code': 67764297, 'gens': [1, 3, 4], 'pres': [5, 2, 2, 2, 2, 2, 10, 243, 58]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101743012388990, 91655865498367938]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 122297461254, 30594431879, 149885733128, 61113682502]}, 'Perm': {'d': 12, 'gens': [43908489, 5887, 16, 87091216, 11520]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2.C_4^2', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
