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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '288.139', 'ambient_counter': 139, 'ambient_order': 288, 'ambient_tex': 'C_2^3.D_{18}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 288, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 6, 'label': '288.139.1.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '1.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': False, 'quotient': '1.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_1', 'simple': False, 'solvable': True, 'special_labels': ['R', 'L0', 'D0', 'C0'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '288.139', 'subgroup_hash': 139, 'subgroup_order': 288, 'subgroup_tex': 'C_2^3.D_{18}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '288.139', 'aut_centralizer_order': None, 'aut_label': '1.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '72.a1.a1', 'complements': ['288.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '2.b1.a1', '2.c1.a1', '2.d1.a1', '2.d1.b1', '2.e1.a1', '2.e1.b1', '3.a1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6608, 4942, 4441, 4041, 4416, 3751, 4247, 3814], 'generators': [1, 32, 144, 2, 96, 72, 4], 'label': '288.139.1.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': [], 'normal_contains': ['2.a1.a1', '2.b1.a1', '2.c1.a1', '2.d1.b1', '2.d1.a1', '2.e1.b1', '2.e1.a1'], 'normalizer': '1.a1.a1', 'old_label': '1.a1.a1', 'projective_image': '72.17', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1.a1.a1', 'subgroup_fusion': None, 'weyl_group': '72.17'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 36, 'aut_gen_orders': [18, 12, 6, 12, 6, 6], 'aut_gens': [[1, 2, 4, 8], [273, 146, 4, 200], [145, 150, 148, 190], [101, 2, 4, 152], [85, 2, 148, 94], [229, 6, 4, 236], [49, 2, 148, 250]], 'aut_group': None, 'aut_hash': 5611529273695941998, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6912, 'aut_permdeg': 44, 'aut_perms': [132002457199141263861217600141170788564022203610115689, 1113605701872301320153655893334537403559116064753620037, 1460840548851290619922678614846169090336153850870747847, 1997665155815639362337831257274431828369122075614840683, 486713947900015347599856405973230579784592592959249076, 465862830868801390264533421433811474593169953522202108], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 36, 1, 1], [3, 2, 1, 1], [4, 2, 4, 1], [4, 36, 1, 1], [4, 36, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [9, 2, 3, 1], [12, 2, 8, 1], [18, 2, 3, 1], [18, 2, 6, 1], [18, 2, 12, 1], [36, 2, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_9.(C_2^4\\times C_6).C_2^3', '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': 18, 'autcentquo_group': '108.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 108, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{18}:C_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 36, 1], [3, 2, 1], [4, 2, 4], [4, 36, 3], [6, 2, 7], [9, 2, 3], [12, 2, 8], [18, 2, 21], [36, 2, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '72.17', 'commutator_count': 1, 'commutator_label': '36.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 139, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 36, 1, 1], [3, 2, 1, 1], [4, 2, 2, 2], [4, 36, 1, 3], [6, 2, 1, 3], [6, 2, 2, 2], [9, 2, 3, 1], [12, 2, 4, 2], [18, 2, 3, 3], [18, 2, 6, 2], [36, 2, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1008, 'exponent': 36, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '12.5', 'frattini_quotient': '24.14', 'hash': 139, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 2, 1, 18], 'inner_gens': [[1, 146, 4, 140], [145, 2, 4, 8], [1, 2, 4, 8], [165, 2, 4, 8]], 'inner_hash': 17, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'C_2\\times D_{18}', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 70]], 'label': '288.139', 'linC_count': 864, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 16, 'linQ_dim': 12, 'linQ_dim_count': 16, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.D18', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 78, 'number_divisions': 28, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 78, 'number_subgroup_classes': 117, 'number_subgroups': 484, 'old_label': None, 'order': 288, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 43], [3, 2], [4, 116], [6, 14], [9, 6], [12, 16], [18, 42], [36, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 6, 4, 8], [1, 146, 4, 8], [1, 2, 4, 152], [1, 150, 148, 106]], 'outer_group': '96.167', 'outer_hash': 167, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [373680, 1537920, 27349344, 4737844], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 4], [6, 4], [8, 2], [12, 2], [24, 2]], 'representations': {'PC': {'code': 2872660390844913022314986499, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, -2, 2, -2, -3, -3, 2045, 3923, 80, 9524, 102, 10757, 166, 9414]}, 'GLZN': {'d': 2, 'p': 72, 'gens': [13252967, 374977, 13810213, 16474276, 373825, 14094019, 20812465]}, 'Perm': {'d': 21, 'gens': [2452841514600838615, 371696065925184000, 5366321611486214400, 1037836800, 7689253435928064000, 57658, 104227]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_2^3.D_{18}', 'transitive_degree': 144, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 36, 'aut_gen_orders': [18, 12, 6, 12, 6, 6], 'aut_gens': [[1, 2, 4, 8], [273, 146, 4, 200], [145, 150, 148, 190], [101, 2, 4, 152], [85, 2, 148, 94], [229, 6, 4, 236], [49, 2, 148, 250]], 'aut_group': None, 'aut_hash': 5611529273695941998, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6912, 'aut_permdeg': 44, 'aut_perms': [132002457199141263861217600141170788564022203610115689, 1113605701872301320153655893334537403559116064753620037, 1460840548851290619922678614846169090336153850870747847, 1997665155815639362337831257274431828369122075614840683, 486713947900015347599856405973230579784592592959249076, 465862830868801390264533421433811474593169953522202108], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 36, 1, 1], [3, 2, 1, 1], [4, 2, 4, 1], [4, 36, 1, 1], [4, 36, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [9, 2, 3, 1], [12, 2, 8, 1], [18, 2, 3, 1], [18, 2, 6, 1], [18, 2, 12, 1], [36, 2, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_9.(C_2^4\\times C_6).C_2^3', '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': 18, 'autcentquo_group': '108.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 108, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{18}:C_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 36, 1], [3, 2, 1], [4, 2, 4], [4, 36, 3], [6, 2, 7], [9, 2, 3], [12, 2, 8], [18, 2, 21], [36, 2, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '72.17', 'commutator_count': 1, 'commutator_label': '36.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 139, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 36, 1, 1], [3, 2, 1, 1], [4, 2, 2, 2], [4, 36, 1, 3], [6, 2, 1, 3], [6, 2, 2, 2], [9, 2, 3, 1], [12, 2, 4, 2], [18, 2, 3, 3], [18, 2, 6, 2], [36, 2, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1008, 'exponent': 36, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '12.5', 'frattini_quotient': '24.14', 'hash': 139, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 2, 1, 18], 'inner_gens': [[1, 146, 4, 140], [145, 2, 4, 8], [1, 2, 4, 8], [165, 2, 4, 8]], 'inner_hash': 17, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'C_2\\times D_{18}', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 70]], 'label': '288.139', 'linC_count': 864, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 16, 'linQ_dim': 12, 'linQ_dim_count': 16, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.D18', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 78, 'number_divisions': 28, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 78, 'number_subgroup_classes': 117, 'number_subgroups': 484, 'old_label': None, 'order': 288, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 43], [3, 2], [4, 116], [6, 14], [9, 6], [12, 16], [18, 42], [36, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 6, 4, 8], [1, 146, 4, 8], [1, 2, 4, 152], [1, 150, 148, 106]], 'outer_group': '96.167', 'outer_hash': 167, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [373680, 1537920, 27349344, 4737844], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 4], [6, 4], [8, 2], [12, 2], [24, 2]], 'representations': {'PC': {'code': 2872660390844913022314986499, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, -2, 2, -2, -3, -3, 2045, 3923, 80, 9524, 102, 10757, 166, 9414]}, 'GLZN': {'d': 2, 'p': 72, 'gens': [13252967, 374977, 13810213, 16474276, 373825, 14094019, 20812465]}, 'Perm': {'d': 21, 'gens': [2452841514600838615, 371696065925184000, 5366321611486214400, 1037836800, 7689253435928064000, 57658, 104227]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_2^3.D_{18}', 'transitive_degree': 144, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, 'wreath_data': None, 'wreath_product': False}
