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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '576.3648', 'ambient_counter': 3648, 'ambient_order': 576, 'ambient_tex': 'C_6.(D_4\\times C_{12})', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 36, 'counter': 65, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '576.3648.8.p1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '8.p1.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': 8, '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': '72.48', 'subgroup_hash': 48, 'subgroup_order': 72, 'subgroup_tex': 'C_6\\times D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '576.3648', 'aut_centralizer_order': 64, 'aut_label': '8.p1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '48.51', 'aut_weyl_index': 128, 'centralizer': '24.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1'], 'contains': ['16.e1.a1', '16.i1.a1', '16.n1.a1', '16.n1.b1', '16.s1.a1', '24.bg1.a1', '24.bh1.a1'], 'core': '16.e1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [8305, -1, 4983, -1, 1575, -1, 7227, -1], 'generators': [1, 192, 28, 288, 16], 'label': '576.3648.8.p1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.a1', 'old_label': '8.p1.a1', 'projective_image': '48.39', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.p1.a1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 3, 3, 2, 2], 'aut_gens': [[1, 2, 12], [1, 10, 60], [1, 10, 13], [1, 10, 12], [36, 39, 13], [1, 26, 12], [1, 38, 12], [1, 39, 12]], 'aut_group': '288.1028', 'aut_hash': 1028, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 9, 'aut_perms': [7, 5040, 1, 5760, 30, 41040, 90720], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 4, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 6, 1], [6, 2, 3, 1], [6, 2, 6, 1], [6, 3, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6\\times S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '48.48', 'autcent_hash': 48, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [3, 1, 2], [3, 2, 3], [6, 1, 6], [6, 2, 9], [6, 3, 8]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 48, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 2, 3], [6, 2, 1, 3], [6, 2, 2, 3], [6, 3, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 364, 'exponent': 6, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '72.48', 'hash': 48, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 3], 'inner_gens': [[1, 2, 12], [1, 2, 60], [1, 26, 12]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 12]], 'label': '72.48', 'linC_count': 96, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 12, 'linQ_dim': 4, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6*D6', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 36, 'number_divisions': 24, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 27, 'number_subgroup_classes': 69, 'number_subgroups': 118, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 15], [3, 8], [6, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[36, 39, 49], [1, 10, 12], [36, 39, 13], [1, 3, 12], [1, 38, 12]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [7, 136, 325, 125, 134], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 4]], 'representations': {'PC': {'code': 201166209349, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -3, -2, -3, 26, 608, 58, 609]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16858573, 7115239, 35931072]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [15191433, 30419024, 36826717, 34603218, 10065962]}, 'Perm': {'d': 10, 'gens': [367921, 806400, 367920, 144, 3]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_6\\times D_6', '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': '48.44', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 'aut_gens': [[1, 2, 8, 48], [5, 2, 8, 48], [25, 2, 8, 48], [289, 2, 8, 48], [1, 6, 8, 48], [1, 26, 8, 48], [1, 290, 8, 48], [1, 2, 8, 52], [1, 2, 8, 72], [1, 2, 8, 336], [1, 2, 40, 48], [1, 2, 8, 240], [193, 2, 8, 48]], 'aut_group': None, 'aut_hash': 9221721648667112671, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6144, 'aut_permdeg': 42, 'aut_perms': [29578474701426762013380040328111375943859775785822, 28884687798886589202968939634324436187139331474150, 31781083637609416227215153232954542052630949691250, 364121001044519936504706367809471249264319982438644, 530680432480486964812016950960611901811310162117196, 1302981292944873369595033565858203580728413720057936, 103635667523865342578534161823229128570160, 175967703315093892541736592114252412128560, 62303075533903889048565625575665755374000, 78731141886604247040000, 24498402866084397389228808689324520198763641149775, 25070442675186907933712164044305532441256513527374], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 12, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 2, 4, 1], [4, 4, 2, 2], [4, 6, 4, 2], [4, 12, 2, 1], [6, 1, 2, 7], [6, 2, 1, 7], [6, 2, 2, 7], [6, 12, 4, 1], [12, 2, 8, 1], [12, 4, 4, 5], [12, 4, 8, 3], [12, 6, 8, 2], [12, 12, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^{10}\\times S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '1024.djt', 'autcent_hash': 1957374279625110839, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 12, 2], [3, 1, 2], [3, 2, 3], [4, 2, 4], [4, 4, 4], [4, 6, 8], [4, 12, 2], [6, 1, 14], [6, 2, 21], [6, 12, 4], [12, 2, 8], [12, 4, 44], [12, 6, 16], [12, 12, 4]], 'center_label': '24.15', 'center_order': 24, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '12.5', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3648, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['192.228', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 12, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 2, 2, 2], [4, 4, 2, 2], [4, 6, 2, 4], [4, 12, 1, 2], [6, 1, 2, 7], [6, 2, 1, 7], [6, 2, 2, 7], [6, 12, 2, 2], [12, 2, 4, 2], [12, 4, 2, 2], [12, 4, 4, 10], [12, 6, 4, 4], [12, 12, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 8736, 'exponent': 12, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '72.48', 'hash': 3648, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 1, 6], 'inner_gens': [[1, 290, 8, 264], [289, 2, 8, 72], [1, 2, 8, 48], [409, 26, 8, 48]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48], [2, 84], [4, 12]], 'label': '576.3648', 'linC_count': 13312, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1248, 'linQ_dim': 10, 'linQ_dim_count': 1008, 'linR_count': 720, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6.(D4*C12)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 52, 'number_characteristic_subgroups': 102, 'number_conjugacy_classes': 144, 'number_divisions': 66, 'number_normal_subgroups': 134, 'number_subgroup_autclasses': 332, 'number_subgroup_classes': 418, 'number_subgroups': 1188, 'old_label': None, 'order': 576, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 31], [3, 8], [4, 96], [6, 104], [12, 336]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[5, 2, 8, 48], [25, 2, 8, 48], [289, 2, 8, 48], [1, 6, 8, 48], [1, 290, 8, 48], [1, 2, 8, 52], [1, 2, 8, 336], [1, 2, 40, 48]], 'outer_group': '256.56092', 'outer_hash': 56092, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 16, 'outer_perms': [1307674368000, 6227020800, 39916800, 362880, 5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 22], [8, 16]], 'representations': {'PC': {'code': 5115954612596306644792817105000524297, 'gens': [1, 2, 4, 6], 'pres': [8, 2, 2, 2, 2, 3, 2, 2, 3, 4641, 41, 91, 12677, 1741, 141, 26886, 166, 24583]}, 'GLZN': {'d': 2, 'p': 56, 'gens': [1268701, 175625, 1580553, 2329683, 177185, 5092893, 2327723, 7551531]}, 'Perm': {'d': 18, 'gens': [355687428464056, 753226703636400, 1321126369920, 325, 6266937600, 1174320, 356995103638320, 435]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6.(D_4\\times C_{12})', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
