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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '864.4378', 'ambient_counter': 4378, 'ambient_order': 864, 'ambient_tex': 'C_6.D_6^2', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 12, 'counter': 294, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '864.4378.18.r1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '18.r1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 18, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '48.37', 'subgroup_hash': 37, 'subgroup_order': 48, 'subgroup_tex': 'D_{12}:C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '864.4378', 'aut_centralizer_order': 8, 'aut_label': '18.r1', 'aut_quo_index': None, 'aut_stab_index': 18, 'aut_weyl_group': '48.51', 'aut_weyl_index': 144, 'centralizer': '216.i1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.o1.b1', '6.t1.b1', '9.a1.a1'], 'contains': ['36.x1.b1', '36.y1.b1', '36.bc1.b1', '36.bp1.a1', '36.bt1.a1', '36.cj1.a1', '36.co1.b1', '54.g1.b1'], 'core': '72.e1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [1114, -1, 9831, -1, 8267, -1, 5182, -1], 'generators': [433, 288, 438, 468, 72], 'label': '864.4378.18.r1.b1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '2.g1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '18.r1.b1', 'projective_image': '432.759', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.r1.b1', 'subgroup_fusion': None, 'weyl_group': '24.14'}
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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': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 3], 'aut_gens': [[26365, 182, 15381], [2209, 182, 24173], [2209, 182, 15381], [26365, 182, 13193], [26365, 949, 13193], [26365, 2184, 15381], [26365, 1560, 15381]], 'aut_group': '96.209', 'aut_hash': 209, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 9, 'aut_perms': [5047, 120, 23, 121, 7, 45360], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 2, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 6, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [12, 2, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 6, 2], [3, 2, 1], [4, 1, 2], [4, 2, 1], [4, 6, 2], [6, 2, 3], [12, 2, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 37, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 2], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 6, 1, 2], [6, 2, 1, 1], [6, 2, 2, 1], [12, 2, 2, 2]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 336, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '24.14', 'hash': 37, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 6], 'inner_gens': [[26365, 2184, 15381], [2209, 182, 4401], [26365, 806, 15381]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'D_6', 'inner_used': [1, 2, 3], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 10]], 'label': '48.37', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D12:C2', 'ngens': 3, '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': 12, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 18, 'number_divisions': 14, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 40, 'number_subgroups': 76, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 15], [3, 2], [4, 16], [6, 6], [12, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [26365, 2198, 2198], 'outer_gens': [[26365, 1417, 15381], [26365, 182, 13193], [26365, 182, 24173]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 1], [8, 1]], 'representations': {'PC': {'code': 67674052929, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -2, 2, -3, 26, 883, 58, 804]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [108378803883894673, 24992906201428620, 41624321546194544]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [26365, 182, 15381]}, 'Perm': {'d': 11, 'gens': [3719545, 8443680, 1499064, 12501360, 3]}}, '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': 'D_{12}:C_2', 'transitive_degree': 24, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [4, 6, 6, 12, 12, 4, 12], 'aut_gens': [[1, 2, 12, 144], [462, 465, 692, 720], [77, 34, 660, 216], [77, 130, 660, 792], [438, 457, 332, 216], [510, 489, 404, 144], [462, 557, 328, 720], [462, 461, 332, 216]], 'aut_group': None, 'aut_hash': 9142008714284746175, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6912, 'aut_permdeg': 20, 'aut_perms': [101647673170433234, 534640888299507987, 532737014516362101, 500055971387239475, 12222784367855000, 98843934761367409, 553178666874865722], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 2, 1], [2, 18, 2, 1], [2, 27, 2, 1], [2, 54, 1, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [4, 3, 2, 1], [4, 6, 1, 1], [4, 6, 2, 1], [4, 9, 2, 1], [4, 18, 1, 1], [4, 18, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 1, 2], [6, 4, 2, 3], [6, 8, 1, 1], [6, 8, 2, 2], [6, 12, 2, 2], [6, 24, 2, 1], [6, 36, 2, 1], [12, 6, 4, 1], [12, 12, 2, 5], [12, 18, 2, 1], [12, 24, 2, 1], [12, 36, 1, 1], [12, 36, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3.C_2^6.C_2^2', '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': 12, 'autcentquo_group': '432.741', 'autcentquo_hash': 741, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^3:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 6, 2], [2, 18, 2], [2, 27, 2], [2, 54, 1], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 3, 2], [4, 6, 3], [4, 9, 2], [4, 18, 3], [6, 2, 3], [6, 4, 8], [6, 8, 5], [6, 12, 4], [6, 24, 2], [6, 36, 2], [12, 6, 4], [12, 12, 10], [12, 18, 2], [12, 24, 2], [12, 36, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '432.759', 'commutator_count': 1, 'commutator_label': '54.15', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4378, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 2], [2, 18, 1, 2], [2, 27, 1, 2], [2, 54, 1, 1], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 3, 2, 1], [4, 6, 1, 3], [4, 9, 2, 1], [4, 18, 1, 3], [6, 2, 1, 3], [6, 4, 1, 8], [6, 8, 1, 5], [6, 12, 1, 4], [6, 24, 1, 2], [6, 36, 1, 2], [12, 6, 2, 2], [12, 12, 1, 6], [12, 12, 2, 2], [12, 18, 2, 1], [12, 24, 1, 2], [12, 36, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 22145760, 'exponent': 12, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '432.759', 'hash': 4378, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 6, 6], 'inner_gens': [[1, 10, 12, 216], [5, 2, 60, 216], [1, 98, 12, 720], [73, 74, 300, 144]], 'inner_hash': 759, 'inner_nilpotent': False, 'inner_order': 432, 'inner_split': False, 'inner_tex': 'C_2\\times S_3^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 28], [4, 22], [8, 6]], 'label': '864.4378', 'linC_count': 32, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6.D6^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 42, 'number_characteristic_subgroups': 46, 'number_conjugacy_classes': 72, 'number_divisions': 65, 'number_normal_subgroups': 142, 'number_subgroup_autclasses': 486, 'number_subgroup_classes': 850, 'number_subgroups': 6448, 'old_label': None, 'order': 864, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 159], [3, 26], [4, 96], [6, 246], [12, 336]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 432], 'outer_gens': [[73, 10, 12, 720], [1, 82, 60, 144], [1, 2, 132, 144], [438, 457, 112, 144]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 24], [4, 14], [8, 11]], 'representations': {'PC': {'code': 1318215911692528847135520678673725982342666437909285, 'gens': [1, 2, 4, 7], 'pres': [8, 2, 2, 3, 2, 2, 3, 2, 3, 576, 161, 41, 194, 971, 91, 2412, 116, 2317, 12102, 6062, 3390, 166, 3103]}, 'Perm': {'d': 17, 'gens': [21009968179336, 44549148268943, 65384759870640, 89295478272000, 21010450809600, 45360, 325, 435]}}, 'schur_multiplier': [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_6.D_6^2', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
