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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '2000.957', 'ambient_counter': 957, 'ambient_order': 2000, 'ambient_tex': 'C_5\\times D_{10}^2', 'central': False, 'central_factor': False, 'centralizer_order': 500, 'characteristic': True, 'core_order': 100, 'counter': 28, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '2000.957.20.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '20.e1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '20.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_{10}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '100.16', 'subgroup_hash': 16, 'subgroup_order': 100, 'subgroup_tex': 'C_{10}^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2000.957', 'aut_centralizer_order': 1600, 'aut_label': '20.e1', 'aut_quo_index': 3, 'aut_stab_index': 1, 'aut_weyl_group': '192.379', 'aut_weyl_index': 1600, 'centralizer': '4.c1', 'complements': ['100.m1'], 'conjugacy_class_count': 1, 'contained_in': ['4.c1', '10.c1', '10.d1'], 'contains': ['40.b1', '100.c1', '100.k1'], 'core': '20.e1', 'coset_action_label': None, 'count': 1, 'diagramx': [7373, 7307, 7237, 6930], 'generators': [1, 40, 400, 1000], 'label': '2000.957.20.e1', 'mobius_quo': 2, 'mobius_sub': -2, 'normal_closure': '20.e1', 'normal_contained_in': ['4.c1', '10.c1', '10.d1'], 'normal_contains': ['40.b1', '100.c1'], 'normalizer': '1.a1', 'old_label': '20.e1', 'projective_image': '500.50', 'quotient_action_image': '4.2', 'quotient_action_kernel': '5.1', 'quotient_action_kernel_order': 5, 'quotient_fusion': None, 'short_label': '20.e1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '100.16', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 120, 'aut_gen_orders': [4, 2, 24, 12, 24, 2], 'aut_gens': [[1, 10], [3, 30], [9, 90], [75, 9], [52, 75], [47, 99], [1, 15]], 'aut_group': '2880.e', 'aut_hash': 6470342117996083924, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2880, 'aut_permdeg': 27, 'aut_perms': [2985219234912708067956516960, 404584072208770564331061864, 6671996747517596363201027523, 4549913727857456859416236588, 7943942149660559119292950561, 404584072208770564331061865], 'aut_phi_ratio': 72.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 24, 1], [10, 1, 72, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times \\GL(2,5)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 120, 'autcent_group': '2880.e', 'autcent_hash': 6470342117996083924, 'autcent_nilpotent': False, 'autcent_order': 2880, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_3\\times \\GL(2,5)', '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], [5, 1, 24], [10, 1, 72]], 'center_label': '100.16', 'center_order': 100, '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', '5.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 16, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 6], [10, 1, 4, 18]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 10, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '100.16', 'hash': 16, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 10], [1, 10]], '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, 100]], 'label': '100.16', 'linC_count': 1440, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 90, 'linQ_dim': 8, 'linQ_dim_count': 90, 'linR_count': 360, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C10^2', 'ngens': 4, 'nilpotency_class': 1, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 100, 'number_divisions': 28, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 40, 'number_subgroups': 40, 'old_label': None, 'order': 100, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [5, 24], [10, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [4, 2, 24, 12, 24, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[3, 30], [9, 90], [75, 9], [52, 75], [47, 99], [1, 15]], 'outer_group': '2880.e', 'outer_hash': 6470342117996083924, 'outer_nilpotent': False, 'outer_order': 2880, 'outer_permdeg': 27, 'outer_perms': [2985219234912708067956516960, 404584072208770564331061864, 6671996747517596363201027523, 4549913727857456859416236588, 7943942149660559119292950561, 404584072208770564331061865], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'S_3\\times \\GL(2,5)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 24]], 'representations': {'PC': {'code': 14419251, 'gens': [1, 3], 'pres': [4, -2, -5, -2, -5, 8, 34]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [7988, 2663]}, 'Perm': {'d': 14, 'gens': [6227020800, 39916800, 1451520, 96]}}, 'schur_multiplier': [10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10, 10], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}^2', 'transitive_degree': 100, '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': '80.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [24, 4, 4, 20, 8], 'aut_gens': [[1, 2, 20, 200], [1000, 1508, 1251, 1161], [1000, 174, 1980, 801], [1, 1058, 580, 601], [1, 505, 1571, 1160], [1, 1516, 930, 1160]], 'aut_group': None, 'aut_hash': 8722775065876318396, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 307200, 'aut_permdeg': 44, 'aut_perms': [962148219039828842963334944855537681298483665799729796, 365362778520272523675110263266798981310745562646478153, 2165114092581844623131856414170058563748286594666109344, 2100960933401701483116393847407732459421241276280548503, 623240759688524795741408837615686643861753256209278624], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 5, 8, 1], [2, 25, 4, 1], [5, 1, 4, 1], [5, 2, 4, 1], [5, 2, 16, 1], [5, 4, 4, 1], [5, 4, 16, 1], [10, 1, 12, 1], [10, 2, 12, 1], [10, 2, 48, 1], [10, 4, 12, 1], [10, 4, 48, 1], [10, 5, 32, 1], [10, 10, 16, 1], [10, 10, 64, 1], [10, 25, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{10}^2.A_4.C_4^3.C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '384.20079', 'autcent_hash': 20079, 'autcent_nilpotent': False, 'autcent_order': 384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^5.D_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 40, 'autcentquo_group': '800.1191', 'autcentquo_hash': 1191, 'autcentquo_nilpotent': False, 'autcentquo_order': 800, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 5, 8], [2, 25, 4], [5, 1, 4], [5, 2, 20], [5, 4, 20], [10, 1, 12], [10, 2, 60], [10, 4, 60], [10, 5, 32], [10, 10, 80], [10, 25, 16]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '100.13', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 957, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 2], ['2.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 5, 1, 8], [2, 25, 1, 4], [5, 1, 4, 1], [5, 2, 2, 2], [5, 2, 4, 4], [5, 4, 2, 2], [5, 4, 4, 4], [10, 1, 4, 3], [10, 2, 2, 6], [10, 2, 4, 12], [10, 4, 2, 6], [10, 4, 4, 12], [10, 5, 4, 8], [10, 10, 2, 8], [10, 10, 4, 16], [10, 25, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 15741180, 'exponent': 10, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '2000.957', 'hash': 957, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [1, 2, 10, 5], 'inner_gens': [[1, 2, 20, 200], [1, 2, 180, 200], [1, 42, 20, 1800], [1, 2, 420, 200]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 100, 'inner_split': False, 'inner_tex': 'D_5^2', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 160], [4, 80]], 'label': '2000.957', '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': 'C5*D10^2', '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': 18, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 320, 'number_divisions': 104, 'number_normal_subgroups': 208, 'number_subgroup_autclasses': 108, 'number_subgroup_classes': 768, 'number_subgroups': 4696, 'old_label': None, 'order': 2000, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 143], [5, 124], [10, 1732]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 4, 2, 2, 4], 'outer_gen_pows': [0, 102, 102, 0, 102, 2, 0, 0, 0, 0], 'outer_gens': [[1, 3, 21, 1800], [1000, 1003, 1061, 401], [1, 108, 1610, 1160], [1, 1002, 180, 1800], [1, 1003, 1061, 600], [1, 1019, 140, 200], [1, 1002, 1060, 601], [1, 3, 180, 200], [1, 1002, 1180, 200], [1, 14, 20, 1800]], 'outer_group': '3072.bjf', 'outer_hash': 2540948649422516732, 'outer_nilpotent': False, 'outer_order': 3072, 'outer_permdeg': 16, 'outer_perms': [1334625465600, 9522120343680, 9434463051000, 8028672825600, 8027634993960, 1334142846720, 6919178112000, 3052245369600, 3052800576000, 5169], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\wr D_6.C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [4, 32], [8, 40], [16, 16]], 'representations': {'PC': {'code': 422494996277553603030236206367539, 'gens': [1, 2, 4, 6], 'pres': [7, -2, -2, -5, -2, -5, -2, -5, 36, 2530, 80, 2811, 3806, 124, 3947]}, 'GLZN': {'d': 2, 'p': 55, 'gens': [2662016, 166391, 166981, 166409, 3493896, 7336769, 5656784]}, 'Perm': {'d': 19, 'gens': [6490430090323207, 6450777815328016, 23, 16, 46200, 13919253929683200, 20742699656448000]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_{10}^2', 'transitive_degree': 80, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1, 2], [10, 13], [10, 15]], 'aut_group': '24.5', 'aut_hash': 5, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [127, 857], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 4, 1], [10, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.5', 'autcent_hash': 5, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times 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], [5, 1, 4], [10, 1, 12]], 'center_label': '20.5', 'center_order': 20, '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', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 1], [10, 1, 4, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '20.5', 'hash': 5, '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, 20]], 'label': '20.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 6, 'linQ_dim': 5, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C10', 'ngens': 3, 'nilpotency_class': 1, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 20, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [5, 4], [10, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[10, 13], [10, 15]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [127, 857], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4]], 'representations': {'PC': {'code': 2179, 'gens': [1, 2], 'pres': [3, -2, -2, -5, 16]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706303489327]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [13320, 4001]}, 'Perm': {'d': 9, 'gens': [40320, 720, 96]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{10}', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
