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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1344.8601', 'ambient_counter': 8601, 'ambient_order': 1344, 'ambient_tex': 'D_{168}:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 84, 'counter': 113, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.8601.8.bk1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '8.bk1.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': '168.35', 'subgroup_hash': 35, 'subgroup_order': 168, 'subgroup_tex': 'C_4\\times D_{21}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.8601', 'aut_centralizer_order': 8, 'aut_label': '8.bk1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '1008.906', 'aut_weyl_index': 16, 'centralizer': '336.i1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.x1.a1', '4.bn1.a1', '4.bq1.a1'], 'contains': ['16.g1.a1', '16.j1.a1', '16.l1.a1', '24.bd1.a1', '56.be1.a1'], 'core': '16.g1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [2843, -1, 7701, -1, 2336, -1, 1854, -1], 'generators': [1177, 672, 448, 6, 8], 'label': '1344.8601.8.bk1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.x1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.h1.a1', 'old_label': '8.bk1.a1', 'projective_image': '672.1150', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.bk1.a1', 'subgroup_fusion': None, 'weyl_group': '168.50'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [2, 2, 6, 42], 'aut_gens': [[1, 2], [85, 2], [1, 58], [1, 34], [65, 86]], 'aut_group': '1008.906', 'aut_hash': 906, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1008, 'aut_permdeg': 14, 'aut_perms': [7, 127, 2405934863, 14530119856], 'aut_phi_ratio': 21.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 21, 2, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 21, 2, 1], [6, 2, 1, 1], [7, 2, 3, 1], [12, 2, 2, 1], [14, 2, 3, 1], [21, 2, 6, 1], [28, 2, 6, 1], [42, 2, 6, 1], [84, 2, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_6\\times F_7', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 21, 2], [3, 2, 1], [4, 1, 2], [4, 21, 2], [6, 2, 1], [7, 2, 3], [12, 2, 2], [14, 2, 3], [21, 2, 6], [28, 2, 6], [42, 2, 6], [84, 2, 12]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '42.5', 'commutator_count': 1, 'commutator_label': '21.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 35, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['42.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 21, 1, 2], [3, 2, 1, 1], [4, 1, 2, 1], [4, 21, 2, 1], [6, 2, 1, 1], [7, 2, 3, 1], [12, 2, 2, 1], [14, 2, 3, 1], [21, 2, 6, 1], [28, 2, 6, 1], [42, 2, 6, 1], [84, 2, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 84, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[2, 0, 12]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '84.14', 'hash': 35, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 21], 'inner_gens': [[1, 82], [89, 2]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 42, 'inner_split': True, 'inner_tex': 'D_{21}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 40]], 'label': '168.35', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 10, 'linQ_dim': 10, 'linQ_dim_count': 10, 'linR_count': 30, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4*D21', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 14, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 48, 'number_divisions': 15, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 32, 'number_subgroups': 172, 'old_label': None, 'order': 168, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 43], [3, 2], [4, 44], [6, 2], [7, 6], [12, 4], [14, 6], [21, 12], [28, 12], [42, 12], [84, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 58], [85, 2], [1, 94]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [24, 40320, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1], [6, 2], [12, 3], [24, 1]], 'representations': {'PC': {'code': 14545626073034338476875, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -3, -7, 821, 26, 2462, 42, 3203, 78, 3604]}, 'GLFp': {'d': 2, 'p': 41, 'gens': [2756841, 314966]}, 'Perm': {'d': 14, 'gens': [479001727, 4032000, 7983360, 6706022400, 973]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times D_{21}', 'transitive_degree': 84, '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': 2, 'aut_exponent': 84, 'aut_gen_orders': [14, 42, 6, 6, 6, 12, 6, 12], 'aut_gens': [[1, 2, 4, 56], [673, 50, 452, 616], [1, 722, 228, 56], [673, 706, 932, 280], [673, 674, 36, 728], [673, 10, 1156, 616], [1009, 10, 484, 728], [1, 34, 940, 1064], [337, 690, 948, 56]], 'aut_group': None, 'aut_hash': 7058334312241270196, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 28, 'aut_perms': [243345853252340087519863774815, 3774282231624467458264434246, 240797245517136651348618753659, 239975150048568325330635176723, 240378245124405025440417961765, 185949619026466208800972495239, 3774571247245376080775944057, 131892487615188812707274315066], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [2, 6, 1, 1], [2, 14, 1, 1], [2, 28, 1, 1], [2, 42, 1, 1], [2, 84, 1, 2], [3, 2, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 6, 1, 1], [4, 12, 1, 2], [4, 14, 1, 1], [4, 28, 1, 1], [4, 42, 1, 1], [6, 2, 1, 1], [6, 8, 1, 1], [6, 14, 2, 1], [6, 56, 1, 1], [7, 2, 3, 1], [8, 4, 1, 1], [8, 12, 1, 1], [8, 14, 2, 1], [8, 84, 1, 1], [12, 4, 1, 1], [12, 8, 1, 1], [12, 28, 1, 1], [12, 56, 1, 1], [14, 2, 3, 1], [14, 8, 3, 1], [14, 12, 3, 1], [21, 4, 3, 1], [24, 4, 2, 1], [24, 28, 2, 1], [28, 4, 3, 1], [28, 8, 3, 1], [28, 12, 3, 1], [28, 24, 3, 2], [42, 4, 3, 1], [42, 16, 3, 1], [56, 8, 3, 1], [56, 24, 3, 1], [84, 8, 3, 1], [84, 16, 3, 1], [168, 8, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '1008.906', 'autcentquo_hash': 906, 'autcentquo_nilpotent': False, 'autcentquo_order': 1008, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 1], [2, 6, 1], [2, 14, 1], [2, 28, 1], [2, 42, 1], [2, 84, 2], [3, 2, 1], [4, 2, 1], [4, 4, 1], [4, 6, 1], [4, 12, 2], [4, 14, 1], [4, 28, 1], [4, 42, 1], [6, 2, 1], [6, 8, 1], [6, 14, 2], [6, 56, 1], [7, 2, 3], [8, 4, 1], [8, 12, 1], [8, 14, 2], [8, 84, 1], [12, 4, 1], [12, 8, 1], [12, 28, 1], [12, 56, 1], [14, 2, 3], [14, 8, 3], [14, 12, 3], [21, 4, 3], [24, 4, 2], [24, 28, 2], [28, 4, 3], [28, 8, 3], [28, 12, 3], [28, 24, 6], [42, 4, 3], [42, 16, 3], [56, 8, 3], [56, 24, 3], [84, 8, 3], [84, 16, 3], [168, 8, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '672.1150', 'commutator_count': 1, 'commutator_label': '84.6', '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': 8601, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [2, 6, 1, 1], [2, 14, 1, 1], [2, 28, 1, 1], [2, 42, 1, 1], [2, 84, 1, 2], [3, 2, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 6, 1, 1], [4, 12, 1, 2], [4, 14, 1, 1], [4, 28, 1, 1], [4, 42, 1, 1], [6, 2, 1, 1], [6, 8, 1, 1], [6, 14, 2, 1], [6, 56, 1, 1], [7, 2, 3, 1], [8, 4, 1, 1], [8, 12, 1, 1], [8, 14, 2, 1], [8, 84, 1, 1], [12, 4, 1, 1], [12, 8, 1, 1], [12, 28, 1, 1], [12, 56, 1, 1], [14, 2, 3, 1], [14, 8, 3, 1], [14, 12, 3, 1], [21, 4, 3, 1], [24, 4, 2, 1], [24, 28, 2, 1], [28, 4, 3, 1], [28, 8, 3, 1], [28, 12, 3, 1], [28, 24, 3, 2], [42, 4, 3, 1], [42, 16, 3, 1], [56, 8, 3, 1], [56, 24, 3, 1], [84, 8, 3, 1], [84, 16, 3, 1], [168, 8, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 59754240, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, 1, 6]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '336.219', 'hash': 8601, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 14, 12], 'inner_gens': [[1, 2, 676, 1064], [1, 2, 724, 56], [673, 682, 4, 280], [337, 2, 1124, 56]], 'inner_hash': 1150, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': True, 'inner_tex': 'D_{12}:D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 36], [4, 26], [8, 12]], 'label': '1344.8601', 'linC_count': 96, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 64, 'linQ_dim': 16, 'linQ_dim_count': 64, 'linR_count': 6, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D168:C2^2', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 49, 'number_characteristic_subgroups': 126, 'number_conjugacy_classes': 90, 'number_divisions': 49, 'number_normal_subgroups': 126, 'number_subgroup_autclasses': 516, 'number_subgroup_classes': 516, 'number_subgroups': 4868, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 263], [3, 2], [4, 120], [6, 94], [7, 6], [8, 128], [12, 96], [14, 66], [21, 12], [24, 64], [28, 216], [42, 60], [56, 96], [84, 72], [168, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 2, 4, 952], [1, 674, 4, 56], [1, 2, 20, 56]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [720, 40320, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 2], [6, 8], [8, 1], [12, 6], [16, 1], [24, 2], [48, 1]], 'representations': {'PC': {'code': 2007692088955095678563930124923123758443624356612017489, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -7, -2, -2, -2, -3, 16226, 8698, 66, 779, 42564, 2820, 116, 37637, 6741, 141, 15702, 166, 14359]}, 'Perm': {'d': 26, 'gens': [621574835364359849165663, 356042890382842, 3074066620654, 1349476938624, 4491510688614, 5634020044188, 6758061133824000, 16754357281155028254720000]}}, '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': 'D_{168}:C_2^2', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
