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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1024.dhl', 'ambient_counter': 2222, 'ambient_order': 1024, 'ambient_tex': '(D_4\\times C_2^3).Q_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 1, 'counter': 289, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1024.dhl.64.bz2.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '64.bz2.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': 64, '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': '16.14', 'subgroup_hash': 14, 'subgroup_order': 16, 'subgroup_tex': 'C_2^4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1024.dhl', 'aut_centralizer_order': 64, 'aut_label': '64.bz2', 'aut_quo_index': None, 'aut_stab_index': 16, 'aut_weyl_group': '8.3', 'aut_weyl_index': 1024, 'centralizer': '32.l1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['32.l1.a1', '32.bt1.a1', '32.bu1.a1'], 'contains': ['128.bb2.a1', '128.bb4.b1', '128.bf1.a1', '128.bj3.a1', '128.by1.a2', '128.by2.b1'], 'core': '1024.a1.a1', 'coset_action_label': None, 'count': 8, 'diagramx': [7605, -1, 6136, -1, 7614, -1, 6140, -1], 'generators': [1313901388807, 6354403201, 5167, 40284847], 'label': '1024.dhl.64.bz2.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '16.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '8.j1.b1', 'old_label': '64.bz2.a1', 'projective_image': '1024.dhl', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '64.bz2.a1', 'subgroup_fusion': None, 'weyl_group': '4.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 420, 'aut_gen_orders': [6, 3], 'aut_gens': [[1, 2, 4, 8], [12, 4, 5, 3], [10, 14, 7, 5]], 'aut_group': '20160.a', 'aut_hash': 3764836782182912467, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20160, 'aut_permdeg': 8, 'aut_perms': [5193, 5760], 'aut_phi_ratio': 2520.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 15, 1]], 'aut_supersolvable': False, 'aut_tex': 'A_8', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 420, 'autcent_group': '20160.a', 'autcent_hash': 3764836782182912467, 'autcent_nilpotent': False, 'autcent_order': 20160, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'A_8', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 15]], 'center_label': '16.14', 'center_order': 16, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 4]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '16.14', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16]], 'label': '16.14', 'linC_count': 840, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 840, 'linQ_dim': 4, 'linQ_dim_count': 840, 'linR_count': 840, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 16, 'number_divisions': 16, 'number_normal_subgroups': 67, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 67, 'number_subgroups': 67, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 420, 'outer_gen_orders': [6, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[12, 4, 5, 3], [10, 14, 7, 5]], 'outer_group': '20160.a', 'outer_hash': 3764836782182912467, 'outer_nilpotent': False, 'outer_order': 20160, 'outer_permdeg': 8, 'outer_perms': [5193, 5760], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'A_8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4], 'pres': [4, -2, 2, 2, 2]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7115648, 35812976, 7115160]}, 'GLFp': {'d': 4, 'p': 2, 'gens': [33837, 18465, 27183, 18467]}, 'Perm': {'d': 8, 'gens': [5040, 120, 6, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4', 'transitive_degree': 16, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [8, 8, 4, 4, 2, 4, 2, 2, 2], 'aut_gens': [[5606713341487, 19614899428423], [6920487710647, 175811776], [5606713336327, 1314164643263], [15309936265552, 2789881493167], [8402518656142, 16825107030359], [5612940725046, 19614939708103], [6920614725127, 19608712329457], [5606673782407, 20916387055303], [5606713341487, 20916346780783], [6920575171200, 19614899428423]], 'aut_group': '8192.bbt', 'aut_hash': 2144635052841457525, 'aut_nilpotency_class': 6, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 384, 'aut_perms': 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1], [4, 8, 2, 2], [4, 16, 1, 1], [4, 32, 2, 1], [4, 64, 1, 1], [4, 64, 2, 2], [8, 32, 8, 1], [8, 64, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^5:C_4.D_4^2', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': None, 'autcentquo_hash': 9209672534681614391, 'autcentquo_nilpotent': True, 'autcentquo_order': 2048, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^5.C_2.C_2^5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 3], [2, 8, 2], [2, 16, 1], [2, 32, 1], [4, 8, 4], [4, 16, 1], [4, 32, 2], [4, 64, 5], [8, 32, 8], [8, 64, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '512.1899', 'commutator_count': 1, 'commutator_label': '128.513', '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', '2.1', '2.1', '2.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 2222, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 3], [2, 8, 1, 2], [2, 16, 1, 1], [2, 32, 1, 1], [4, 8, 1, 4], [4, 16, 1, 1], [4, 32, 2, 1], [4, 64, 1, 3], [4, 64, 2, 1], [8, 32, 2, 4], [8, 64, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 48, 'exponent': 8, 'exponents_of_order': [10], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[8, 0, 4], [8, 1, 4]], 'familial': False, 'frattini_label': '256.1534', 'frattini_quotient': '4.2', 'hash': 5502138377298815999, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [8, 8], 'inner_gens': [[5606713341487, 302803223], [11212509393342, 19614899428423]], 'inner_hash': 5284139924001968831, 'inner_nilpotent': True, 'inner_order': 512, 'inner_split': False, 'inner_tex': 'C_2^3.C_2\\wr C_4', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 10], [4, 5], [8, 10], [16, 1]], 'label': '1024.dhl', 'linC_count': 8, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': '(D4*C2^3).Q16', 'ngens': 2, 'nilpotency_class': 7, 'nilpotent': True, 'normal_counts': [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': 18, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 34, 'number_divisions': 25, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 319, 'number_subgroup_classes': 478, 'number_subgroups': 4949, 'old_label': None, 'order': 1024, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 79], [4, 432], [8, 512]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [6266942646, 0, 6266942641, 0], 'outer_gens': [[8396292003256, 16818880372558], [8396331557183, 20916346775623], [15309849525832, 4103869598640], [5606713341487, 20916387060456]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [6, 415, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 7], [8, 7], [16, 3]], 'representations': {'PC': {'code': '30106282878508839298391796064394657236571552034267932583306596143952263136698135', 'gens': [1, 2, 5, 7, 8, 9], 'pres': [10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 8000, 3641, 51, 29642, 82, 23363, 7204, 21614, 1824, 334, 144, 15375, 40326, 3376, 10666, 29457, 3227, 32418, 10108, 8678, 268, 1619]}, 'Perm': {'d': 16, 'gens': [5606713341487, 19614899428423]}}, 'schur_multiplier': [2, 2], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(D_4\\times C_2^3).Q_{16}', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
