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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1344.11633', 'ambient_counter': 11633, 'ambient_order': 1344, 'ambient_tex': 'C_{84}.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 672, 'characteristic': False, 'core_order': 12, 'counter': 116, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1344.11633.112.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '112.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '112.42', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 42, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 112, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2\\times D_{14}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '12.5', 'subgroup_hash': 5, 'subgroup_order': 12, 'subgroup_tex': 'C_2\\times C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.11633', 'aut_centralizer_order': None, 'aut_label': '112.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2.a1', 'complements': [], 'conjugacy_class_count': 5, 'contained_in': ['16.a1', '56.a1', '56.b1', '56.d1', '56.e1', '56.f1'], 'contains': ['224.a1', '224.b1', '336.a1'], 'core': '112.a1', 'coset_action_label': None, 'count': 5, 'diagramx': None, 'generators': [681, 16, 448], 'label': '1344.11633.112.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '112.a1', 'normal_contained_in': ['16.a1', '56.a1', '56.b1'], 'normal_contains': ['224.a1', '336.a1'], 'normalizer': '1.a1', 'old_label': '112.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '112.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [6, 9], [6, 11]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 31], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [6, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.4', 'autcent_hash': 4, 'autcent_nilpotent': False, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_6', '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], [3, 1, 2], [6, 1, 6]], 'center_label': '12.5', 'center_order': 12, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [6, 1, 2, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '12.5', '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, 12]], 'label': '12.5', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 6, 'linQ_dim': 3, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C6', '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': 12, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 9], [6, 11]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4]], 'representations': {'PC': {'code': 273, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16507]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 2064]}, 'Perm': {'d': 7, 'gens': [720, 24, 4]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], '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_6', 'transitive_degree': 12, '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': '96.231', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[1, 2, 4, 8, 32], [17, 2, 4, 8, 416], [1, 18, 4, 8, 416], [1, 2, 20, 8, 416], [1, 2, 4, 24, 416], [1, 2, 4, 8, 432], [1, 2, 4, 8, 1312], [1, 2, 12, 8, 1312], [681, 2, 684, 8, 1312], [1, 2, 4, 9, 1333], [1, 2, 4, 8, 608], [1, 194, 4, 8, 32]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 322560, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 5, 1], [2, 7, 2, 1], [2, 14, 5, 1], [3, 1, 2, 1], [4, 2, 10, 1], [4, 14, 10, 1], [6, 1, 2, 1], [6, 2, 10, 1], [6, 7, 4, 1], [6, 14, 10, 1], [7, 2, 3, 1], [12, 2, 20, 1], [12, 14, 20, 1], [14, 2, 3, 1], [14, 4, 15, 1], [21, 2, 6, 1], [28, 4, 30, 1], [42, 2, 6, 1], [42, 4, 30, 1], [84, 4, 60, 1]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 5], [2, 7, 2], [2, 14, 5], [3, 1, 2], [4, 2, 10], [4, 14, 10], [6, 1, 2], [6, 2, 10], [6, 7, 4], [6, 14, 10], [7, 2, 3], [12, 2, 20], [12, 14, 20], [14, 2, 3], [14, 4, 15], [21, 2, 6], [28, 4, 30], [42, 2, 6], [42, 4, 30], [84, 4, 60]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '224.196', 'commutator_count': 1, 'commutator_label': '14.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 11633, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['3.1', 1], ['32.50', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 5], [2, 7, 1, 2], [2, 14, 1, 5], [3, 1, 2, 1], [4, 2, 1, 10], [4, 14, 1, 10], [6, 1, 2, 1], [6, 2, 2, 5], [6, 7, 2, 2], [6, 14, 2, 5], [7, 2, 3, 1], [12, 2, 2, 10], [12, 14, 2, 10], [14, 2, 3, 1], [14, 4, 3, 5], [21, 2, 6, 1], [28, 4, 3, 10], [42, 2, 6, 1], [42, 4, 6, 5], [84, 4, 6, 10]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4033075200, 'exponent': 84, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, 0, 6]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '672.1277', 'hash': 11633, 'hyperelementary': 2, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 224, 'inner_split': None, 'inner_tex': 'C_2^3\\times D_{14}', 'inner_used': None, 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 16, 'irrep_stats': [[1, 96], [2, 144], [4, 6], [8, 9]], 'label': '1344.11633', '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': 'C84.C2^4', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 255, 'number_divisions': 102, 'number_normal_subgroups': 890, 'number_subgroup_autclasses': 148, 'number_subgroup_classes': 1588, 'number_subgroups': 5736, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 95], [3, 2], [4, 160], [6, 190], [7, 6], [12, 320], [14, 66], [21, 12], [28, 120], [42, 132], [84, 240]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': True, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '1440.5852', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 1440, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': '\\GL(2,4):C_2^3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2, 3], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 32], [4, 2], [6, 16], [8, 2], [12, 16], [24, 1], [48, 1]], 'representations': {'PC': {'code': 25807253105412518181067089905386364411046923, 'gens': [1, 2, 3, 4, 6], 'pres': [8, -2, -2, -2, -2, -2, 2, -3, -7, 114, 211, 91, 2309, 9997, 141, 23310, 222, 36879]}, 'Perm': {'d': 26, 'gens': [621574835364359896320598, 3054901973058, 4496735693276, 5901854882697, 7302781821506, 6758061133824000, 1334672640598, 16754357281155028254720000]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{84}.C_2^4', 'transitive_degree': 336, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [4, 2, 3, 2, 4, 2, 7, 2], 'aut_gens': [[1, 2, 4, 8], [58, 2, 5, 67], [1, 2, 4, 104], [1, 2, 4, 88], [1, 2, 60, 8], [57, 56, 62, 66], [1, 2, 7, 8], [1, 2, 100, 8], [1, 2, 5, 8]], 'aut_group': '56448.n', 'aut_hash': 7440696882083854005, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 56448, 'aut_permdeg': 15, 'aut_perms': [635323409280, 538, 435, 88216168320, 32619646080, 288369849600, 3676, 193158120960], 'aut_phi_ratio': 1176.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [2, 7, 8, 1], [7, 2, 3, 1], [14, 2, 21, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_7\\times C_2^3:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '1344.11686', 'autcent_hash': 11686, 'autcent_nilpotent': False, 'autcent_order': 1344, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3:\\GL(3,2)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 7, 8], [7, 2, 3], [14, 2, 21]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 42, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 7, 1, 8], [7, 2, 3, 1], [14, 2, 3, 7]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 855, 'exponent': 14, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '112.42', 'hash': 42, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [1, 1, 2, 7], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 104], [1, 2, 20, 8]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 24]], 'label': '112.42', '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': 'C2^2*D14', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 40, 'number_divisions': 24, 'number_normal_subgroups': 83, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 134, 'number_subgroups': 440, 'old_label': None, 'order': 112, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 63], [7, 6], [14, 42]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 84, 'outer_gen_orders': [6, 21], 'outer_gen_pows': [0, 4], 'outer_gens': [[1, 3, 60, 24], [58, 59, 63, 82]], 'outer_group': '4032.cj', 'outer_hash': 8242728398078336135, 'outer_nilpotent': False, 'outer_order': 4032, 'outer_permdeg': 11, 'outer_perms': [8731468, 1461844], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_3\\times C_2^3:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [6, 8]], 'representations': {'PC': {'code': 8076441809669, 'gens': [1, 2, 3, 4], 'pres': [5, -2, -2, -2, -2, -7, 533, 58, 614]}, 'GLZN': {'d': 2, 'p': 21, 'gens': [9325, 120406, 9269, 74096, 185354]}, 'Perm': {'d': 13, 'gens': [40284846, 288, 1, 415, 559566000]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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': 'C_2^2\\times D_{14}', 'transitive_degree': 56, 'wreath_data': None, 'wreath_product': False}
