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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1024.dfb', 'ambient_counter': 2160, 'ambient_order': 1024, 'ambient_tex': 'C_2^6.\\OD_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 4, 'counter': 37, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1024.dfb.16.q1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '16.q1', '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': 16, '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': '64.202', 'subgroup_hash': 202, 'subgroup_order': 64, 'subgroup_tex': 'C_2^3:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1024.dfb', 'aut_centralizer_order': 16, 'aut_label': '16.q1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '128.2151', 'aut_weyl_index': 128, 'centralizer': '64.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.a1'], 'contains': ['32.n1', '32.o1', '32.q1', '32.v1', '32.w1', '32.y1', '32.ba1', '32.bd1', '32.bf1'], 'core': '256.a1', 'coset_action_label': None, 'count': 8, 'diagramx': [3073, -1, 3017, -1], 'generators': [5606275029696, 6227388847, 40279687, 126, 5167, 1313941673647], 'label': '1024.dfb.16.q1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '8.a1', 'old_label': '16.q1', 'projective_image': '512.1753', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.q1', 'subgroup_fusion': None, 'weyl_group': '8.5'}
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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': 4, 'aut_exponent': 24, 'aut_gen_orders': [8, 4], 'aut_gens': [[4001, 643, 2565, 1571, 3993], [517, 2727, 3623, 545, 687], [1957, 1575, 3623, 1571, 1561]], 'aut_group': '12288.dp', 'aut_hash': 1630168457023225010, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 24, 'aut_perms': [187522706853152502407387, 128367454260810836992011], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 1, 4, 1], [2, 2, 12, 1], [2, 4, 2, 1], [4, 4, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^9.S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 8092891664598236742, 'autcent_nilpotent': True, 'autcent_order': 2048, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8.C_2^3', '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, 2, 12], [2, 4, 2], [4, 4, 6]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 202, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['32.27', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 12], [2, 4, 1, 2], [4, 4, 1, 6]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 420, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '16.14', 'hash': 202, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 1, 2], 'inner_gens': [[4001, 643, 2565, 1571, 2731], [4001, 643, 2565, 1571, 2699], [4001, 643, 2565, 1571, 3993], [4001, 643, 2565, 1571, 3993], [2691, 1921, 2565, 1571, 3993]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12]], 'label': '64.202', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, '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': 'C2^3:D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 28, 'number_divisions': 28, 'number_normal_subgroups': 105, 'number_subgroup_autclasses': 38, 'number_subgroup_classes': 331, 'number_subgroups': 569, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 39], [4, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [513, 513, 513, 513, 513, 513, 513, 513, 513, 513], 'outer_gens': [[1575, 643, 2565, 545, 3993], [2695, 3619, 2565, 1539, 1593], [2723, 675, 2565, 1571, 1981], [4001, 643, 3623, 1571, 3993], [2723, 1921, 3591, 1571, 3993], [1957, 2695, 2565, 1571, 1981], [1957, 643, 2565, 1571, 3993], [4001, 1921, 2565, 1571, 3993], [4001, 1953, 2565, 1571, 3993], [2723, 643, 2565, 1571, 3993]], 'outer_group': '1536.408632814', 'outer_hash': 2972356385187470528, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 14, 'outer_perms': [1683987865, 7878390024, 21317548327, 14390022966, 21797289414, 243538566, 85686, 2438997120, 21316418640, 2440489806], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12]], 'representations': {'PC': {'code': 4564455696, 'gens': [1, 2, 3, 4, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 724, 730, 88]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [705686951426, 141602720900, 706461793378, 706204576418]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [11040833, 31029334, 19896385, 10743875, 6210625, 19208257]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [643, 2565, 523, 545, 1539, 687]}, 'Perm': {'d': 10, 'gens': [367925, 766087, 85800, 806520, 806416, 806400]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3:D_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': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [4, 8, 8, 4, 2, 4, 4, 2, 4], 'aut_gens': [[11212860971950, 2809471431606, 12526722806830], [19588164047976, 4110879219840, 20889651674856], [16825323109686, 9676790703064, 16825323104761], [14016025860606, 2809431877800, 15329966803326], [16825282825081, 4110918778801, 18126770451961], [11219088360545, 4117105877760, 12520495428065], [18126770457007, 2809471431727, 16825282830127], [11212861362606, 8369115972191, 12526722471726], [16819095726120, 4110879219961, 18132957561000], [16819095726006, 2803204489086, 18132957560886]], 'aut_group': '16384.mu', 'aut_hash': 4243906216982603733, 'aut_nilpotency_class': 7, 'aut_nilpotent': True, 'aut_order': 16384, 'aut_permdeg': 512, 'aut_perms': 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'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 3], [2, 8, 2, 2], [2, 16, 1, 2], [4, 8, 2, 1], [4, 16, 8, 1], [4, 32, 1, 1], [4, 64, 1, 2], [8, 32, 4, 1], [16, 64, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^7.C_2\\wr D_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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': None, 'autcentquo_hash': 2013466141039987713, 'autcentquo_nilpotent': True, 'autcentquo_order': 4096, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^6.C_2^4.C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 3], [2, 8, 4], [2, 16, 2], [4, 8, 2], [4, 16, 8], [4, 32, 1], [4, 64, 2], [8, 32, 4], [16, 64, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '512.1753', 'commutator_count': 1, 'commutator_label': '64.193', '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': 2160, '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, 4], [2, 16, 1, 2], [4, 8, 1, 2], [4, 16, 2, 4], [4, 32, 1, 1], [4, 64, 2, 1], [8, 32, 2, 2], [16, 64, 4, 2]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 24, 'exponent': 16, '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': '256.1535', 'frattini_quotient': '4.2', 'hash': 4903759005473243375, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [8, 4, 8], 'inner_gens': [[11212860971950, 9683057635504, 11212860976865], [16825323104640, 2809471431606, 18126730172160], [12526722811745, 9683057635504, 12526722806830]], 'inner_hash': 1436668582407764448, 'inner_nilpotent': True, 'inner_order': 512, 'inner_split': False, 'inner_tex': 'C_2^4.C_2^2:C_8', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 4], [4, 6], [8, 10], [16, 1]], 'label': '1024.dfb', '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': False, 'metacyclic': False, 'monomial': True, 'name': 'C2^6.OD16', 'ngens': 3, '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': 17, 'number_characteristic_subgroups': 17, 'number_conjugacy_classes': 37, 'number_divisions': 24, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 330, 'number_subgroup_classes': 682, 'number_subgroups': 6665, 'old_label': None, 'order': 1024, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 79], [4, 304], [8, 128], [16, 512]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [121, 0, 1313941668480, 0], 'outer_gens': [[11212861357560, 9676830251945, 12526722466680], [11212860971950, 2809471426681, 12526722806830], [16825282830127, 2809471431727, 18126770457007], [19581977278086, 4110879224886, 19581977283241]], 'outer_group': '32.27', 'outer_hash': 27, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [15966, 21048, 19342, 7277], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\wr C_2', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5], [8, 8], [16, 3]], 'representations': {'PC': {'code': '29759160622346366286547587349104150709157016332178734886503269069516847867099215', 'gens': [1, 4, 6, 7, 9, 10], 'pres': [10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 20, 51, 3862, 39363, 5933, 6823, 113, 8804, 3614, 40325, 16335, 23526, 8416, 12346, 6476, 206, 51207, 24488, 18738, 17308, 4358, 1148]}, 'Perm': {'d': 16, 'gens': [11212860971950, 2809471431606, 12526722806830]}}, 'schur_multiplier': [2, 2], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^6.\\OD_{16}', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
