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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1056.234', 'ambient_counter': 234, 'ambient_order': 1056, 'ambient_tex': 'C_{24}:D_{22}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 132, 'counter': 18, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1056.234.4.i1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '4.i1.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': 4, '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': '264.5', 'subgroup_hash': 5, 'subgroup_order': 264, 'subgroup_tex': 'C_6.D_{22}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1056.234', 'aut_centralizer_order': 4, 'aut_label': '4.i1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': None, 'aut_weyl_index': 8, 'centralizer': '528.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['8.a1.a1', '8.f1.a1', '8.g1.a1', '12.e1.a1', '44.f1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [3671, -1, 6005, -1, 2956, -1, 3167, -1], 'generators': [1, 528, 704, 1014, 96], 'label': '1056.234.4.i1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.a1', 'old_label': '4.i1.a1', 'projective_image': '528.107', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.i1.a1', 'subgroup_fusion': None, 'weyl_group': '264.34'}
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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': 330, 'aut_gen_orders': [30, 30, 15, 30, 10], 'aut_gens': [[1, 2, 8], [53, 178, 152], [245, 182, 248], [97, 90, 128], [169, 182, 248], [101, 182, 112]], 'aut_group': None, 'aut_hash': 4845733366138083661, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2640, 'aut_permdeg': 28, 'aut_perms': [53807625267380498438360241839, 281308655732017558920296710886, 161438381546052066233753348673, 224037311461706378554120677819, 138417646878528409378798331516], 'aut_phi_ratio': 33.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 2, 1], [3, 2, 1, 1], [4, 3, 2, 1], [4, 33, 2, 1], [6, 2, 1, 1], [6, 22, 2, 1], [11, 2, 5, 1], [22, 2, 5, 1], [33, 4, 5, 1], [44, 6, 10, 1], [66, 4, 5, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{11}:(C_2^2\\times C_{10}\\times S_3)', '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': 330, 'autcentquo_group': '660.15', 'autcentquo_hash': 15, 'autcentquo_nilpotent': False, 'autcentquo_order': 660, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 11, 2], [3, 2, 1], [4, 3, 2], [4, 33, 2], [6, 2, 1], [6, 22, 2], [11, 2, 5], [22, 2, 5], [33, 4, 5], [44, 6, 10], [66, 4, 5]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '132.5', 'commutator_count': 1, 'commutator_label': '33.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.1', 1], ['22.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 1, 2], [3, 2, 1, 1], [4, 3, 2, 1], [4, 33, 2, 1], [6, 2, 1, 1], [6, 22, 1, 2], [11, 2, 5, 1], [22, 2, 5, 1], [33, 4, 5, 1], [44, 6, 10, 1], [66, 4, 5, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 132, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[4, -1, 5]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '132.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [2, 2, 33], 'inner_gens': [[1, 2, 80], [1, 2, 184], [193, 90, 8]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 132, 'inner_split': True, 'inner_tex': 'S_3\\times D_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 20, 'irrQ_dim': 40, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 24], [4, 10]], 'label': '264.5', 'linC_count': 65, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 14, 'linQ_dim_count': 12, 'linR_count': 80, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6.D22', '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': 13, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 42, 'number_divisions': 15, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 32, 'number_subgroups': 184, 'old_label': None, 'order': 264, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 23], [3, 2], [4, 72], [6, 46], [11, 10], [22, 10], [33, 20], [44, 60], [66, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 10], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 6, 80], [5, 2, 104]], 'outer_group': '20.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 20, 'outer_permdeg': 9, 'outer_perms': [40320, 810], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{10}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [10, 2], [20, 3]], 'representations': {'PC': {'code': 143526118308321000019, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -2, -3, -11, 26, 1603, 1848, 78, 6004]}, 'GLZN': {'d': 2, 'p': 44, 'gens': [1959255, 1790625, 3706977, 85361, 1044979]}, 'Perm': {'d': 18, 'gens': [21010450849927, 137, 7, 840, 397708329755520]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6.D_{22}', 'transitive_degree': 132, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [10, 60, 10, 10, 22, 10, 10], 'aut_gens': [[1, 2, 4], [913, 530, 724], [241, 90, 100], [97, 354, 932], [577, 2, 332], [721, 882, 92], [1009, 530, 428], [673, 706, 284]], 'aut_group': None, 'aut_hash': 5956886757194812668, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21120, 'aut_permdeg': 38, 'aut_perms': [266331215863713235960583880839863312785409366, 1024968708838121452580634609121503549236174, 8794121720420980470858496232119102031817770, 216786771555898282526519154106626743154185249, 435650134534312755403554705431689503306061422, 213888807652531873911883297464802923204654600, 431256716863865616530956684993431709353841854], 'aut_phi_ratio': 66.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 2, 1], [2, 12, 1, 1], [2, 132, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 12, 1, 1], [4, 22, 1, 1], [4, 132, 1, 1], [6, 2, 1, 1], [6, 22, 2, 1], [8, 2, 2, 1], [8, 22, 2, 1], [11, 2, 5, 1], [12, 2, 2, 1], [12, 22, 2, 1], [22, 2, 5, 1], [22, 24, 5, 1], [24, 2, 4, 1], [24, 22, 4, 1], [33, 4, 5, 1], [44, 4, 5, 1], [44, 24, 5, 1], [66, 4, 5, 1], [88, 4, 10, 1], [132, 4, 10, 1], [264, 4, 20, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{66}.C_{10}.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': None, 'autcentquo_hash': 4845733366138083661, 'autcentquo_nilpotent': False, 'autcentquo_order': 2640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{11}:(C_2^2\\times C_{10}\\times S_3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 11, 2], [2, 12, 1], [2, 132, 1], [3, 2, 1], [4, 2, 1], [4, 12, 1], [4, 22, 1], [4, 132, 1], [6, 2, 1], [6, 22, 2], [8, 2, 2], [8, 22, 2], [11, 2, 5], [12, 2, 2], [12, 22, 2], [22, 2, 5], [22, 24, 5], [24, 2, 4], [24, 22, 4], [33, 4, 5], [44, 4, 5], [44, 24, 5], [66, 4, 5], [88, 4, 10], [132, 4, 10], [264, 4, 20]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '528.107', 'commutator_count': 1, 'commutator_label': '132.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 234, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['22.1', 1], ['48.6', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 1, 2], [2, 12, 1, 1], [2, 132, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 12, 1, 1], [4, 22, 1, 1], [4, 132, 1, 1], [6, 2, 1, 1], [6, 22, 1, 2], [8, 2, 2, 1], [8, 22, 2, 1], [11, 2, 5, 1], [12, 2, 2, 1], [12, 22, 2, 1], [22, 2, 5, 1], [22, 24, 5, 1], [24, 2, 4, 1], [24, 22, 4, 1], [33, 4, 5, 1], [44, 4, 5, 1], [44, 24, 5, 1], [66, 4, 5, 1], [88, 4, 10, 1], [132, 4, 10, 1], [264, 4, 20, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 16128, 'exponent': 264, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[4, 0, 20]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '264.34', 'hash': 234, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 132, 'inner_gen_orders': [2, 2, 132], 'inner_gens': [[1, 2, 964], [1, 2, 620], [97, 442, 4]], 'inner_hash': 107, 'inner_nilpotent': False, 'inner_order': 528, 'inner_split': True, 'inner_tex': 'D_{11}\\times D_{12}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 42], [4, 55]], 'label': '1056.234', 'linC_count': 180, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 32, 'linQ_dim': 16, 'linQ_dim_count': 32, 'linR_count': 80, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C24:D22', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 28, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 105, 'number_divisions': 30, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 128, 'number_subgroup_classes': 136, 'number_subgroups': 1808, 'old_label': None, 'order': 1056, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 167], [3, 2], [4, 168], [6, 46], [8, 48], [11, 10], [12, 48], [22, 130], [24, 96], [33, 20], [44, 140], [66, 20], [88, 40], [132, 40], [264, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 10], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 2, 260], [529, 2, 964], [1, 2, 628]], 'outer_group': '40.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 40, 'outer_permdeg': 11, 'outer_perms': [720, 3628800, 40353], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{10}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 4], [8, 2], [10, 4], [20, 3], [40, 2], [80, 1]], 'representations': {'PC': {'code': 46518043711503110055521243574627999032625724760000019, 'gens': [1, 2, 3], 'pres': [7, -2, -2, -2, -2, -2, -3, -11, 20246, 6519, 58, 24419, 2586, 80, 24084, 6451, 102, 13445, 15468, 166, 47046]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [12413213, 6612431, 433456, 287893, 6806977, 291149, 13131557]}, 'Perm': {'d': 22, 'gens': [51347038427613296047, 506875508305689600, 109743842217993984000, 163498485693134592000, 51347038427612928000, 6706022400, 4364893]}}, 'schur_multiplier': [2, 2], '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_{24}:D_{22}', 'transitive_degree': 264, 'wreath_data': None, 'wreath_product': False}
