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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '432.258', 'ambient_counter': 258, 'ambient_order': 432, 'ambient_tex': 'C_3^2:\\GL(2,3)', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 72, 'counter': 4, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '432.258.3.b1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '3.b1.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': 3, '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': '144.122', 'subgroup_hash': 122, 'subgroup_order': 144, 'subgroup_tex': 'C_3\\times \\GL(2,3)', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '432.258', 'aut_centralizer_order': 1, 'aut_label': '3.b1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '96.226', 'aut_weyl_index': 9, 'centralizer': '72.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.b1.a1', '9.a1.a1', '9.c1.a1', '12.b1.a1'], 'core': '6.b1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [4902, -1, 7125, -1, 3359, -1, 5368, -1], 'generators': [1, 216, 324, 228, 2, 18], 'label': '432.258.3.b1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '3.b1.a1', 'projective_image': '72.43', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3.b1.a1', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', '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, 6, 2, 2], 'aut_gens': [[54880, 48561, 121487], [54880, 88999, 121487], [54880, 28709, 9213], [82320, 6260, 5148], [54880, 116439, 5148], [54880, 116439, 46239]], 'aut_group': '96.226', 'aut_hash': 226, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [16, 127, 1447, 11520, 5160], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 12, 1, 1], [3, 1, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [4, 6, 1, 1], [6, 1, 2, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 12, 2, 1], [8, 6, 2, 1], [12, 6, 2, 1], [24, 6, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times S_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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 12, 1], [3, 1, 2], [3, 8, 3], [4, 6, 1], [6, 1, 2], [6, 8, 3], [6, 12, 2], [8, 6, 2], [12, 6, 2], [24, 6, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '24.12', 'commutator_count': 1, 'commutator_label': '24.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 122, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['48.29', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 12, 1, 1], [3, 1, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [4, 6, 1, 1], [6, 1, 2, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 12, 2, 1], [8, 6, 2, 1], [12, 6, 2, 1], [24, 6, 4, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 72, 'exponent': 24, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 4], [4, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '72.42', 'hash': 122, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [1, 4, 3], 'inner_gens': [[54880, 48561, 121487], [54880, 48561, 125552], [54880, 88152, 121487]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'S_4', 'inner_used': [2, 3], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 6], [2, 9], [3, 6], [4, 3]], 'label': '144.122', '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': False, 'metacyclic': False, 'monomial': False, 'name': 'C3*GL(2,3)', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_conjugacy_classes': 24, 'number_divisions': 14, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 33, 'number_subgroup_classes': 35, 'number_subgroups': 128, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 13], [3, 26], [4, 6], [6, 50], [8, 12], [12, 12], [24, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [6860, 6860], 'outer_gens': [[82320, 78107, 121487], [54880, 88999, 121487]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [3, 2], [4, 3], [6, 2], [8, 2]], 'representations': {'PC': {'code': 51443752244301380996775697, 'gens': [1, 2, 3, 4], 'pres': [6, -2, -3, -2, 2, -2, -3, 49, 1406, 1034, 230, 435, 1521, 351, 69, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101728301564804, 108470311056907516]}, 'GLFp': {'d': 2, 'p': 19, 'gens': [54880, 48561, 121487]}, 'Perm': {'d': 11, 'gens': [364560, 3, 1222104, 4578600, 8799984, 12822720]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times \\GL(2,3)', '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': '2.1', '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, 6, 6, 6], 'aut_gens': [[1, 2, 6, 36], [217, 2, 6, 36], [399, 124, 132, 78], [431, 332, 138, 294], [97, 134, 6, 252], [79, 296, 222, 252]], 'aut_group': '864.4000', 'aut_hash': 4000, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 36, 'aut_perms': [350790295161904880829255192511886331661506, 278060451642058136651540454263569757008825, 298016918710292965762685813623385167784327, 288343594600194185945491686220521434167377, 144975578960286951652765711174305988685886], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 36, 1, 1], [3, 1, 2, 1], [3, 6, 1, 1], [3, 24, 3, 1], [4, 6, 1, 1], [6, 1, 2, 1], [6, 6, 1, 1], [6, 24, 3, 1], [6, 36, 2, 1], [8, 18, 2, 1], [12, 6, 2, 1], [12, 12, 1, 1], [12, 12, 2, 1], [24, 18, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_6^2:D_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '432.523', 'autcentquo_hash': 523, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2:D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 36, 1], [3, 1, 2], [3, 6, 1], [3, 24, 3], [4, 6, 1], [6, 1, 2], [6, 6, 1], [6, 24, 3], [6, 36, 2], [8, 18, 2], [12, 6, 2], [12, 12, 3], [24, 18, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': False, 'central_quotient': '72.43', 'commutator_count': 2, 'commutator_label': '216.42', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 258, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 36, 1, 1], [3, 1, 2, 1], [3, 6, 1, 1], [3, 24, 1, 3], [4, 6, 1, 1], [6, 1, 2, 1], [6, 6, 1, 1], [6, 24, 1, 3], [6, 36, 2, 1], [8, 18, 2, 1], [12, 6, 2, 1], [12, 12, 1, 1], [12, 12, 2, 1], [24, 18, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 45360, 'exponent': 24, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 4]], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '72.43', 'hash': 258, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3, 2, 6], 'inner_gens': [[1, 4, 222, 210], [5, 2, 330, 390], [217, 110, 6, 252], [295, 332, 222, 36]], 'inner_hash': 43, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'C_3:S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 6, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 2], [2, 6], [3, 10], [4, 4], [6, 7]], 'label': '432.258', 'linC_count': 16, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 20, 'linQ_dim': 10, 'linQ_dim_count': 20, 'linR_count': 20, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C3^2:GL(2,3)', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 29, 'number_divisions': 20, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 55, 'number_subgroup_classes': 86, 'number_subgroups': 616, 'old_label': None, 'order': 432, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 37], [3, 80], [4, 6], [6, 152], [8, 36], [12, 48], [24, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 4, 246, 42], [1, 430, 222, 426]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 4], [3, 2], [4, 5], [6, 5], [12, 1], [24, 1]], 'representations': {'PC': {'code': 1265858672629836958024183625165490123271274025, 'gens': [1, 2, 3, 5], 'pres': [7, 2, 3, 2, 3, 2, 2, 3, 57, 4664, 3474, 58, 528, 7354, 6836, 1488, 102, 16133, 2028, 124, 4122, 9127]}, 'Perm': {'d': 17, 'gens': [4284310429134, 25300505624123, 44822654916480, 68287549478400, 12460, 18523, 5329]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:\\GL(2,3)', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
