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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1152.156208', 'ambient_counter': 156208, 'ambient_order': 1152, 'ambient_tex': 'Q_8:S_3\\times S_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 4, 'counter': 528, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1152.156208.144.bc1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '144.bc1.a1', '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': 144, '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': '8.3', 'subgroup_hash': 3, 'subgroup_order': 8, 'subgroup_tex': 'D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1152.156208', 'aut_centralizer_order': 32, 'aut_label': '144.bc1', 'aut_quo_index': None, 'aut_stab_index': 18, 'aut_weyl_group': '8.3', 'aut_weyl_index': 576, 'centralizer': '144.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['48.bl1.a1', '48.bn1.a1', '48.bo1.a1', '72.r1.a1', '72.bs1.a1', '72.bt1.a1', '72.ch1.a1', '72.ci1.a1', '72.cj1.a1', '72.ck1.a1'], 'contains': ['288.a1.a1', '288.g1.a1'], 'core': '288.a1.a1', 'coset_action_label': None, 'count': 18, 'diagramx': [7887, -1, 8614, -1, 7843, -1, 2985, -1], 'generators': [3, 72], 'label': '1152.156208.144.bc1.a1', 'mobius_quo': None, 'mobius_sub': -4, 'normal_closure': '4.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.a1.a1', 'old_label': '144.bc1.a1', 'projective_image': '576.8354', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '144.bc1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D4', 'ngens': 2, 'nilpotency_class': 2, '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': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
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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': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 2, 3, 4, 2, 12, 2, 3, 12, 2, 2], 'aut_gens': [[1, 2, 4, 96, 192], [769, 178, 244, 576, 288], [1, 2, 4, 96, 960], [1, 34, 4, 576, 288], [433, 18, 1124, 96, 1056], [1, 674, 100, 96, 192], [385, 722, 1004, 576, 864], [1, 674, 676, 96, 192], [769, 2, 772, 96, 192], [841, 594, 908, 96, 288], [1, 50, 68, 576, 864], [1, 50, 20, 576, 480]], 'aut_group': '4608.lj', 'aut_hash': 3301011005301566777, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4608, 'aut_permdeg': 28, 'aut_perms': [197273428737557526806390628477, 262155610719500861410354874304, 1025131574943895024232, 9696963707909785970062176187, 206969557141431706472, 40916607587535119905139224444, 207174411305130044160, 271031175739979883111562616934, 59129348070151778015447242875, 718841767251114622952, 198914493916266707197703603971], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 2, 1], [2, 12, 1, 1], [2, 36, 1, 1], [2, 72, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 6, 1, 1], [4, 6, 2, 1], [4, 12, 1, 3], [4, 24, 1, 2], [4, 72, 1, 1], [6, 2, 1, 1], [6, 6, 1, 2], [6, 8, 1, 1], [6, 12, 2, 1], [6, 16, 1, 1], [6, 96, 1, 1], [8, 6, 2, 1], [8, 18, 2, 1], [8, 36, 2, 2], [12, 4, 1, 1], [12, 4, 2, 1], [12, 12, 1, 1], [12, 12, 2, 2], [12, 16, 1, 1], [12, 24, 1, 2], [12, 24, 2, 2], [12, 32, 1, 2], [12, 32, 2, 1], [24, 48, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5.D_6^2', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '576.8659', 'autcentquo_hash': 8659, 'autcentquo_nilpotent': False, 'autcentquo_order': 576, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^2:D_6^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [2, 6, 2], [2, 12, 1], [2, 36, 1], [2, 72, 1], [3, 2, 1], [3, 8, 1], [3, 16, 1], [4, 2, 1], [4, 4, 1], [4, 6, 3], [4, 12, 3], [4, 24, 2], [4, 72, 1], [6, 2, 1], [6, 6, 2], [6, 8, 1], [6, 12, 2], [6, 16, 1], [6, 96, 1], [8, 6, 2], [8, 18, 2], [8, 36, 4], [12, 4, 3], [12, 12, 5], [12, 16, 1], [12, 24, 6], [12, 32, 4], [24, 48, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '576.8354', 'commutator_count': 1, 'commutator_label': '144.155', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 156208, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['24.12', 1], ['48.17', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 1, 2], [2, 12, 1, 1], [2, 36, 1, 1], [2, 72, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 6, 1, 3], [4, 12, 1, 3], [4, 24, 1, 2], [4, 72, 1, 1], [6, 2, 1, 1], [6, 6, 1, 2], [6, 8, 1, 1], [6, 12, 1, 2], [6, 16, 1, 1], [6, 96, 1, 1], [8, 6, 2, 1], [8, 18, 2, 1], [8, 36, 2, 2], [12, 4, 1, 1], [12, 4, 2, 1], [12, 12, 1, 3], [12, 12, 2, 1], [12, 16, 1, 1], [12, 24, 1, 2], [12, 24, 2, 2], [12, 32, 1, 2], [12, 32, 2, 1], [24, 48, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 80640, 'exponent': 24, 'exponents_of_order': [7, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 2]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '288.1028', 'hash': 156208, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 12, 2, 6], 'inner_gens': [[1, 2, 76, 96, 960], [1, 2, 68, 576, 864], [25, 34, 4, 576, 1056], [1, 674, 676, 96, 192], [385, 674, 484, 96, 192]], 'inner_hash': 8354, 'inner_nilpotent': False, 'inner_order': 576, 'inner_split': True, 'inner_tex': 'C_2^4:S_3^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 18], [3, 8], [4, 9], [6, 14], [8, 1], [12, 2]], 'label': '1152.156208', 'linC_count': 272, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 16, 'linQ_dim': 7, 'linQ_dim_count': 16, 'linR_count': 16, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'Q8:S3*S4', 'ngens': 9, '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': 46, 'number_characteristic_subgroups': 51, 'number_conjugacy_classes': 60, 'number_divisions': 50, 'number_normal_subgroups': 55, 'number_subgroup_autclasses': 582, 'number_subgroup_classes': 616, 'number_subgroups': 5894, 'old_label': None, 'order': 1152, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 139], [3, 26], [4, 180], [6, 158], [8, 192], [12, 360], [24, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 2, 52, 96, 960], [1, 50, 68, 576, 864], [1, 2, 4, 96, 960]], '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': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [3, 8], [4, 9], [6, 6], [8, 3], [12, 6]], 'representations': {'PC': {'code': '2026260494576056447018483320180857021017861786224143785983584322957211279832317245449', 'gens': [1, 2, 3, 7, 8], 'pres': [9, -2, -2, -2, -2, -2, -3, -2, 2, -3, 2054, 929, 74, 2019, 732, 102, 1813, 130, 1742, 18159, 9096, 5325, 2310, 1374, 69127, 31120, 19033, 7810, 1339, 1996, 214, 62216, 15578]}, 'Perm': {'d': 15, 'gens': [20677628280, 5, 102150720000, 194123381760, 288369849600, 3, 840, 7, 16]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'Q_8:S_3\\times S_4', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
