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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1521.11', 'ambient_counter': 11, 'ambient_order': 1521, 'ambient_tex': 'C_{13}^2:C_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 507, 'characteristic': False, 'core_order': 39, 'counter': 25, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1521.11.39.a1.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '39.a1.e1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '39.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 39, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{13}:C_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '39.2', 'subgroup_hash': 2, 'subgroup_order': 39, 'subgroup_tex': 'C_{39}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1521.11', 'aut_centralizer_order': 79092, 'aut_label': '39.a1', 'aut_quo_index': 1, 'aut_stab_index': 14, 'aut_weyl_group': '24.9', 'aut_weyl_index': 1107288, 'centralizer': '3.a1.a1', 'complements': ['39.b1.bc1', '39.b1.a1', '39.b1.o1', '39.b1.bo1', '39.b1.m1', '39.b1.ba1', '39.b1.bm1', '39.b1.k1', '39.b1.y1', '39.b1.bi1', '39.b1.g1', '39.b1.u1', '39.b1.bp1', '39.b1.n1', '39.b1.bb1', '39.b1.bf1', '39.b1.d1', '39.b1.r1', '39.b1.bk1', '39.b1.i1', '39.b1.w1', '39.b1.be1', '39.b1.c1', '39.b1.q1', '39.b1.bl1', '39.b1.j1', '39.b1.x1', '39.b1.bh1', '39.b1.f1', '39.b1.t1', '39.b1.bj1', '39.b1.h1', '39.b1.v1', '39.b1.bd1', '39.b1.b1', '39.b1.p1', '39.b1.bn1', '39.b1.l1', '39.b1.z1'], 'conjugacy_class_count': 1, 'contained_in': ['3.a1.a1', '13.a1.e1'], 'contains': ['117.a1.e1', '507.a1.a1'], 'core': '39.a1.e1', 'coset_action_label': None, 'count': 1, 'diagramx': [5514, 7699, 212, 5398, 6581, 7244, 3806, 3599], 'generators': [507, 354], 'label': '1521.11.39.a1.e1', 'mobius_quo': 1, 'mobius_sub': 13, 'normal_closure': '39.a1.e1', 'normal_contained_in': ['3.a1.a1'], 'normal_contains': ['117.a1.e1', '507.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '39.a1.e1', 'projective_image': '507.4', 'quotient_action_image': '3.1', 'quotient_action_kernel': '13.1', 'quotient_action_kernel_order': 13, 'quotient_fusion': None, 'short_label': '39.a1.e1', 'subgroup_fusion': None, 'weyl_group': '3.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '39.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1], [14], [28]], 'aut_group': '24.9', 'aut_hash': 9, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 24, 'aut_permdeg': 9, 'aut_perms': [40320, 867], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [13, 1, 12, 1], [39, 1, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{12}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.9', 'autcent_hash': 9, 'autcent_nilpotent': True, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{12}', '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], [3, 1, 2], [13, 1, 12], [39, 1, 24]], 'center_label': '39.2', 'center_order': 39, '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': ['3.1', '13.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['13.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [13, 1, 12, 1], [39, 1, 24, 1]], 'element_repr_type': 'PC', 'elementary': 39, 'eulerian_function': 1, 'exponent': 39, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3, 13], 'faithful_reps': [[1, 0, 24]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '39.2', 'hash': 2, 'hyperelementary': 39, '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': 24, 'irrQ_dim': 24, 'irrR_degree': 2, 'irrep_stats': [[1, 39]], 'label': '39.2', 'linC_count': 24, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 1, 'linQ_dim': 14, 'linQ_dim_count': 1, 'linR_count': 12, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C39', 'ngens': 2, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 39, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 39, 'order_factorization_type': 11, 'order_stats': [[1, 1], [3, 2], [13, 12], [39, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[14], [28]], 'outer_group': '24.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [40320, 867], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{12}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [3, 13], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [12, 1], [24, 1]], 'representations': {'PC': {'code': 743, 'gens': [1], 'pres': [2, -3, -13, 6]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [2211, 19782]}, 'Perm': {'d': 16, 'gens': [2615348736000, 5748019200]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [39], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{39}', 'transitive_degree': 39, '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': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 2184, 'aut_gen_orders': [168, 14, 12], 'aut_gens': [[1, 3, 39], [403, 381, 987], [787, 471, 222], [415, 942, 156]], 'aut_group': None, 'aut_hash': 4205311029593209713, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 26574912, 'aut_permdeg': 507, 'aut_perms': 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'aut_phi_ratio': 28392.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 169, 3, 2], [13, 3, 56, 1], [39, 3, 112, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times C_{13}^2.C_{12}.\\PSL(2,13).C_2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2184, 'autcentquo_group': '4429152.a', 'autcentquo_hash': 6402842459436585890, 'autcentquo_nilpotent': False, 'autcentquo_order': 4429152, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{13}^2.\\GL(2,13)', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 169, 6], [13, 3, 56], [39, 3, 112]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '507.4', 'commutator_count': 1, 'commutator_label': '169.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '13.1', '13.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['507.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 169, 2, 3], [13, 3, 4, 14], [39, 3, 8, 14]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 104, 'exponent': 39, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3, 7, 13], 'factors_of_order': [3, 13], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1521.11', 'hash': 11, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 39, 'inner_gen_orders': [3, 13, 13], 'inner_gens': [[1, 27, 858], [16, 3, 39], [703, 3, 39]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 507, 'inner_split': True, 'inner_tex': 'C_{13}^2:C_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 9], [3, 168]], 'label': '1521.11', 'linC_count': 11648, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 26, 'linQ_degree_count': 273, 'linQ_dim': 26, 'linQ_dim_count': 273, 'linR_count': 2912, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C13^2:C3^2', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 6, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 177, 'number_divisions': 33, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 96, 'number_subgroups': 1440, 'old_label': None, 'order': 1521, 'order_factorization_type': 22, 'order_stats': [[1, 1], [3, 1016], [13, 168], [39, 336]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2184, 'outer_gen_orders': [168, 56, 56], 'outer_gen_pows': [0, 2, 1], 'outer_gens': [[508, 843, 768], [1015, 1068, 690], [1015, 726, 99]], 'outer_group': None, 'outer_hash': 3857937142624469989, 'outer_nilpotent': False, 'outer_order': 52416, 'outer_permdeg': 59, 'outer_perms': [100766027757639723615432562420498812570833679234962280386636146667771980767655827, 35401109161700035017990501229223704015585406074870658990526130860110263331079602, 129044476902660998772823577783750494096628552790354461462495337875536854022475546], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'S_3\\times C_4.\\PSL(2,13).C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 29, 'pgroup': 0, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [12, 14], [24, 14]], 'representations': {'PC': {'code': 299040259044477191, 'gens': [1, 2, 3], 'pres': [4, -3, -13, -3, -13, 217, 10298, 46, 16851]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [4375377422, 7341833547, 8585274507, 10491497597]}, 'Perm': {'d': 29, 'gens': [23053488719126563289755596397, 2615348736000, 337635120120412848224236767610, 611483003307581117509526751610]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{13}^2:C_3^2', 'transitive_degree': 507, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 156, 'aut_gen_orders': [12, 13], 'aut_gens': [[1, 3], [1, 21], [31, 3]], 'aut_group': '156.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 156, 'aut_permdeg': 13, 'aut_perms': [5429374035, 809341288], 'aut_phi_ratio': 6.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 13, 1, 2], [13, 3, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_{13}', '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': 156, 'autcentquo_group': '156.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 156, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{13}', 'cc_stats': [[1, 1, 1], [3, 13, 2], [13, 3, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '39.1', 'commutator_count': 1, 'commutator_label': '13.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '13.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 13, 2, 1], [13, 3, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 8, 'exponent': 39, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [3, 13], 'faithful_reps': [[3, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '39.1', 'hash': 1, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 39, 'inner_gen_orders': [3, 13], 'inner_gens': [[1, 9], [34, 3]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 39, 'inner_split': True, 'inner_tex': 'C_{13}:C_3', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 6, 'irrep_stats': [[1, 3], [3, 4]], 'label': '39.1', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 1, 'linQ_dim': 12, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C13:C3', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 7, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 16, 'old_label': None, 'order': 39, 'order_factorization_type': 11, 'order_stats': [[1, 1], [3, 26], [13, 12]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [1], 'outer_gens': [[1, 33]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [12, 1]], 'representations': {'PC': {'code': 1463, 'gens': [1, 2], 'pres': [2, -3, -13, 37]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [2211, 19776]}, 'Perm': {'d': 13, 'gens': [96855576, 5748019200]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{13}:C_3', 'transitive_degree': 13, 'wreath_data': None, 'wreath_product': False}
