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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1728.47367', 'ambient_counter': 47367, 'ambient_order': 1728, 'ambient_tex': 'Q_8:S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 36, 'counter': 135, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.47367.24.j1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '24.j1', 'outer_equivalence': True, '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': 24, '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': '72.49', 'subgroup_hash': 49, 'subgroup_order': 72, 'subgroup_tex': 'C_6:D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.47367', 'aut_centralizer_order': None, 'aut_label': '24.j1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '216.d1', 'complements': None, 'conjugacy_class_count': 9, 'contained_in': ['8.a1', '12.f1', '12.g1', '12.i1', '12.bb1', '12.bc1'], 'contains': ['48.b1', '48.f1', '48.k1', '48.v1', '72.d1', '72.e1'], 'core': '48.b1', 'coset_action_label': None, 'count': 27, 'diagramx': None, 'generators': [3, 48, 1656, 864, 872], 'label': '1728.47367.24.j1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '24.j1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.j1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [6, 12], 'aut_gens': [[1, 2, 12], [19, 22, 54], [7, 40, 26]], 'aut_group': '10368.qn', 'aut_hash': 3718250448986069408, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10368, 'aut_permdeg': 13, 'aut_perms': [3035622963, 143353458], 'aut_phi_ratio': 432.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 9, 4, 1], [3, 2, 4, 1], [6, 2, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times C_3^2:\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 9, 4], [3, 2, 4], [6, 2, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', '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': 49, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 9, 1, 4], [3, 2, 1, 4], [6, 2, 1, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 7, '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': '72.49', 'hash': 49, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[1, 10, 60], [5, 2, 12], [25, 2, 12]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 16]], 'label': '72.49', 'linC_count': 36, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 36, 'linQ_dim': 4, 'linQ_dim_count': 36, 'linR_count': 36, '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 24, 'number_divisions': 24, 'number_normal_subgroups': 41, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 96, 'number_subgroups': 272, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 39], [3, 8], [6, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 3, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 1, 1], 'outer_gens': [[1, 2, 68], [1, 38, 12], [1, 30, 68], [7, 44, 18], [7, 2, 12], [43, 2, 12], [1, 30, 44], [1, 58, 20]], 'outer_group': '576.8653', 'outer_hash': 8653, 'outer_nilpotent': False, 'outer_order': 576, 'outer_permdeg': 8, 'outer_perms': [720, 2, 1440, 4, 7, 16, 5160, 11520], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16]], 'representations': {'PC': {'code': 1043840168369989, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -3, -2, -3, 101, 26, 122, 1203, 58, 1204]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26483560, 12016555, 35931481]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24822599, 8092916, 16002800, 28816400, 31289712]}, 'Perm': {'d': 10, 'gens': [41046, 721, 1, 30, 403248]}}, 'schur_multiplier': [2, 2, 6], '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_6:D_6', 'transitive_degree': 36, '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [6, 36, 6, 6, 6], 'aut_gens': [[1, 2, 4, 24, 144], [1306, 529, 1356, 224, 1452], [72, 865, 972, 594, 1300], [961, 2, 884, 24, 1020], [1092, 1310, 1312, 1117, 1404], [1224, 913, 60, 1442, 1300]], 'aut_group': None, 'aut_hash': 5213592911207747358, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 248832, 'aut_permdeg': 24, 'aut_perms': [140695305401794826550341, 436273411724512583048318, 308292887249465245683908, 389828652921162357653811, 458908939624996655263818], 'aut_phi_ratio': 432.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 9, 1], [2, 9, 6, 1], [2, 54, 3, 1], [3, 2, 3, 1], [3, 4, 3, 1], [3, 8, 1, 1], [4, 2, 3, 1], [4, 3, 6, 1], [4, 18, 9, 1], [4, 27, 2, 1], [6, 2, 3, 1], [6, 4, 3, 1], [6, 8, 1, 1], [6, 12, 18, 1], [6, 18, 6, 1], [6, 24, 9, 1], [12, 4, 9, 1], [12, 6, 12, 1], [12, 8, 9, 1], [12, 12, 6, 1], [12, 16, 3, 1], [12, 36, 9, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3.C_2^6.D_6^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': '7776.be', 'autcentquo_hash': 4734684673362351397, 'autcentquo_nilpotent': False, 'autcentquo_order': 7776, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^4:S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 9], [2, 9, 6], [2, 54, 3], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 2, 3], [4, 3, 6], [4, 18, 9], [4, 27, 2], [6, 2, 3], [6, 4, 3], [6, 8, 1], [6, 12, 18], [6, 18, 6], [6, 24, 9], [12, 4, 9], [12, 6, 12], [12, 8, 9], [12, 12, 6], [12, 16, 3], [12, 36, 9]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '864.4704', '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', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 47367, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 9], [2, 9, 1, 6], [2, 54, 1, 3], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 2, 1, 3], [4, 3, 2, 3], [4, 18, 1, 9], [4, 27, 2, 1], [6, 2, 1, 3], [6, 4, 1, 3], [6, 8, 1, 1], [6, 12, 1, 18], [6, 18, 1, 6], [6, 24, 1, 9], [12, 4, 1, 9], [12, 6, 2, 6], [12, 8, 1, 9], [12, 12, 2, 3], [12, 16, 1, 3], [12, 36, 1, 9]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 17776640000, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4704', 'hash': 47367, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 6, 6, 6], 'inner_gens': [[1, 2, 4, 120, 1008], [1, 2, 20, 24, 1008], [1, 874, 4, 24, 1008], [49, 2, 4, 24, 1584], [865, 866, 868, 312, 144]], 'inner_hash': 4704, 'inner_nilpotent': False, 'inner_order': 864, 'inner_split': True, 'inner_tex': 'S_3\\times D_6^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 32, 'irrR_degree': 16, 'irrep_stats': [[1, 32], [2, 56], [4, 36], [8, 10], [16, 1]], 'label': '1728.47367', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'Q8:S3^3', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 135, 'number_divisions': 122, 'number_normal_subgroups': 652, 'number_subgroup_autclasses': 418, 'number_subgroup_classes': 4092, 'number_subgroups': 30340, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 271], [3, 26], [4, 240], [6, 566], [12, 624]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 3, 3, 2, 2], 'outer_gen_pows': [432, 3, 0, 0, 73, 0, 0], 'outer_gens': [[433, 434, 1300, 456, 144], [866, 865, 972, 80, 720], [1, 2, 884, 24, 720], [2, 72, 1452, 17, 1392], [433, 434, 436, 552, 300], [865, 2, 868, 120, 144], [865, 2, 20, 888, 720]], 'outer_group': '288.1028', 'outer_hash': 1028, 'outer_nilpotent': False, 'outer_order': 288, 'outer_permdeg': 9, 'outer_perms': [7, 5040, 1, 5760, 30, 41040, 90720], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'D_6\\times S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 48], [4, 28], [8, 10], [16, 4]], 'representations': {'PC': {'code': 19085572680515458943224351694874969803554866086184546594830377, 'gens': [1, 2, 3, 5, 7], 'pres': [9, -2, -2, -2, -3, -2, -3, -2, -2, -3, 281, 74, 15852, 3918, 5404, 130, 5189, 63510, 31767, 15900, 4200, 186, 4363, 214, 3932]}, 'Perm': {'d': 17, 'gens': [21011006021040, 21011006016143, 21011006016136, 44654927788800, 68100219936000, 90603192556800, 325, 45360, 435]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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': 'Q_8:S_3^3', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
