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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '8640.ck', 'ambient_counter': 63, 'ambient_order': 8640, 'ambient_tex': 'A_4\\times D_6\\times A_5', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 12, 'counter': 148, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '8640.ck.60.h1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '60.h1', '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': 60, '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': '144.195', 'subgroup_hash': 195, 'subgroup_order': 144, 'subgroup_tex': 'C_6^2:C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '8640.ck', 'aut_centralizer_order': None, 'aut_label': '60.h1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '360.bm1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['15.b1', '20.b1'], 'contains': ['120.e1', '120.v1', '120.w1', '120.bq1', '120.br1', '180.g1', '180.n1'], 'core': '720.a1', 'coset_action_label': None, 'count': 20, 'diagramx': None, 'generators': [54, 45640552281, 1078479360, 50339126223, 2079342720, 30], 'label': '8640.ck.60.h1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '20.b1', 'old_label': '60.h1', 'projective_image': '4320.ct', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '60.h1', 'subgroup_fusion': None, 'weyl_group': '18.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '48.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [4, 42], 'aut_gens': [[1, 2, 4, 24], [84, 135, 88, 121], [84, 122, 93, 108]], 'aut_group': '16128.bn', 'aut_hash': 4884145202955954088, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 13, 'aut_perms': [2159256984, 247968031], 'aut_phi_ratio': 336.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [2, 3, 8, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 14, 1], [6, 2, 7, 1], [6, 2, 14, 1], [6, 3, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times C_2^4:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '2688.cd', 'autcent_hash': 8270626938983654557, 'autcent_nilpotent': False, 'autcent_order': 2688, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4:\\GL(3,2)', '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, 3, 8], [3, 1, 2], [3, 2, 3], [6, 1, 14], [6, 2, 21], [6, 3, 16]], 'center_label': '24.15', 'center_order': 24, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', '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': 195, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 3, 1, 8], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 2, 7], [6, 2, 1, 7], [6, 2, 2, 7], [6, 3, 2, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 7800, 'exponent': 6, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '144.195', 'hash': 195, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 1, 3], 'inner_gens': [[1, 2, 4, 24], [1, 2, 4, 120], [1, 2, 4, 24], [1, 50, 4, 24]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48], [2, 24]], 'label': '144.195', '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': 'C6^2:C2^2', 'ngens': 6, '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 72, 'number_divisions': 48, 'number_normal_subgroups': 166, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 284, 'number_subgroups': 504, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 31], [3, 8], [6, 104]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 84, 'outer_gen_orders': [3, 2, 2, 7], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 15, 77, 108], [1, 3, 20, 24], [1, 74, 77, 24], [85, 15, 76, 25]], 'outer_group': '2688.cd', 'outer_hash': 8270626938983654557, 'outer_nilpotent': False, 'outer_order': 2688, 'outer_permdeg': 10, 'outer_perms': [2311446, 2805415, 1305528, 2026494], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 24], [4, 8]], 'representations': {'PC': {'code': 203179200087301, 'gens': [1, 2, 3, 5], 'pres': [6, -2, -2, -2, -3, -2, -3, 50, 1810, 88, 1739]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 705557812721, 706200855362, 706589871143]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [19949, 8645, 18685, 1777, 1733, 9587]}, 'Perm': {'d': 12, 'gens': [40723326, 127, 288, 414, 91450800, 130682160]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2:C_2^2', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [2, 2, 12, 2, 3, 2, 6, 3, 2], 'aut_gens': [[680, 30, 1528450560, 766080, 1078116480, 50339126223, 998686080, 649595574], [680, 30, 2079342720, 766080, 1078116480, 50339126223, 124104960, 1448657334], [680, 30, 2403031680, 766080, 1078116480, 50339126223, 998686080, 2079705654], [680, 30, 566133120, 123742080, 998686080, 50339126223, 1078116480, 1528450584], [662, 48, 1528450560, 766080, 1078116480, 50339126241, 998686080, 649595574], [680, 30, 1447891200, 997920000, 124104960, 50339126223, 1078116480, 649595574], [50339123223, 48, 2482462080, 766080, 1078116480, 5033, 998686080, 566496006], [44112105561, 30, 1447891200, 997920000, 124104960, 75247206454, 1078116480, 649595526], [680, 30, 1528450560, 766080, 1078116480, 50339126223, 998686080, 649595526], [680, 30, 1447891200, 766080, 1078116480, 50339126223, 998686080, 2039425974]], 'aut_group': '34560.br', 'aut_hash': 8528631481988113540, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 34560, 'aut_permdeg': 29, 'aut_perms': [54863796223680169549824002861, 294418945596202284192908443721, 2995277998054376033434999476239, 4584633488777800567874167603200, 294418946720203011970516118937, 236945026978604183450355666889, 3654711055628240988329330119285, 3365080110345550385358272102400, 283530075021783204254532763408], 'aut_phi_ratio': 15.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 9, 2, 1], [2, 15, 1, 2], [2, 45, 1, 2], [2, 45, 2, 1], [2, 135, 2, 1], [3, 2, 1, 1], [3, 4, 2, 1], [3, 8, 2, 1], [3, 20, 1, 1], [3, 40, 1, 1], [3, 80, 2, 1], [3, 160, 2, 1], [5, 12, 2, 1], [6, 2, 1, 1], [6, 4, 2, 1], [6, 6, 1, 2], [6, 8, 2, 1], [6, 12, 4, 1], [6, 20, 1, 1], [6, 30, 1, 2], [6, 40, 1, 1], [6, 60, 1, 2], [6, 60, 2, 3], [6, 80, 2, 1], [6, 90, 1, 2], [6, 120, 1, 2], [6, 120, 2, 2], [6, 160, 2, 1], [6, 180, 2, 1], [6, 180, 4, 1], [6, 240, 4, 1], [10, 12, 2, 1], [10, 36, 2, 2], [10, 36, 4, 1], [10, 108, 4, 1], [15, 24, 2, 1], [15, 48, 4, 1], [15, 96, 4, 1], [30, 24, 2, 1], [30, 48, 4, 1], [30, 72, 2, 2], [30, 96, 4, 1], [30, 144, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6\\times S_4\\times S_5', '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': 60, 'autcentquo_group': '17280.bd', 'autcentquo_hash': 456236538737532445, 'autcentquo_nilpotent': False, 'autcentquo_order': 17280, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times S_4\\times S_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 9, 2], [2, 15, 2], [2, 45, 4], [2, 135, 2], [3, 2, 1], [3, 4, 2], [3, 8, 2], [3, 20, 1], [3, 40, 1], [3, 80, 2], [3, 160, 2], [5, 12, 2], [6, 2, 1], [6, 4, 2], [6, 6, 2], [6, 8, 2], [6, 12, 4], [6, 20, 1], [6, 30, 2], [6, 40, 1], [6, 60, 8], [6, 80, 2], [6, 90, 2], [6, 120, 6], [6, 160, 2], [6, 180, 6], [6, 240, 4], [10, 12, 2], [10, 36, 8], [10, 108, 4], [15, 24, 2], [15, 48, 4], [15, 96, 4], [30, 24, 2], [30, 48, 4], [30, 72, 4], [30, 96, 4], [30, 144, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '4320.ct', 'commutator_count': 1, 'commutator_label': '720.772', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '60.5'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 63, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['2.1', 1], ['6.1', 1], ['60.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 9, 1, 2], [2, 15, 1, 2], [2, 45, 1, 4], [2, 135, 1, 2], [3, 2, 1, 1], [3, 4, 2, 1], [3, 8, 2, 1], [3, 20, 1, 1], [3, 40, 1, 1], [3, 80, 2, 1], [3, 160, 2, 1], [5, 12, 2, 1], [6, 2, 1, 1], [6, 4, 2, 1], [6, 6, 1, 2], [6, 8, 2, 1], [6, 12, 2, 2], [6, 20, 1, 1], [6, 30, 1, 2], [6, 40, 1, 1], [6, 60, 1, 4], [6, 60, 2, 2], [6, 80, 2, 1], [6, 90, 1, 2], [6, 120, 1, 2], [6, 120, 2, 2], [6, 160, 2, 1], [6, 180, 1, 2], [6, 180, 2, 2], [6, 240, 2, 2], [10, 12, 2, 1], [10, 36, 2, 4], [10, 108, 2, 2], [15, 24, 2, 1], [15, 48, 4, 1], [15, 96, 4, 1], [30, 24, 2, 1], [30, 48, 4, 1], [30, 72, 2, 2], [30, 96, 4, 1], [30, 144, 4, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 228, 'exponent': 30, 'exponents_of_order': [6, 3, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[18, 1, 2], [24, 1, 1], [30, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8640.ck', 'hash': 6087161459814412481, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [3, 3, 3, 2, 2, 6, 2, 6], 'inner_gens': [[680, 30, 1528450560, 766080, 1078116480, 81474230495, 998686080, 649595526], [680, 30, 1528450560, 766080, 1078116480, 50339126223, 998686080, 649595544], [680, 30, 1528450560, 123742080, 998686080, 50339126223, 124104960, 2079705654], [680, 30, 2482462080, 766080, 1078116480, 50339126223, 998686080, 566496054], [680, 30, 2482462080, 766080, 1078116480, 50339126223, 998686080, 566496054], [44112104840, 30, 1528450560, 766080, 1078116480, 50339126223, 998686080, 649595544], [680, 30, 1447891200, 766080, 1078116480, 50339126223, 998686080, 2039425974], [662, 48, 1447891200, 997920000, 124104960, 50339126241, 1078116480, 649595574]], 'inner_hash': 1199672556059335690, 'inner_nilpotent': False, 'inner_order': 4320, 'inner_split': True, 'inner_tex': 'S_3\\times A_4\\times A_5', 'inner_used': [1, 2, 3, 4, 6, 8], 'irrC_degree': 18, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 18, 'irrep_stats': [[1, 12], [2, 6], [3, 28], [4, 12], [5, 12], [6, 14], [8, 6], [9, 8], [10, 6], [12, 4], [15, 4], [18, 4], [24, 2], [30, 2]], 'label': '8640.ck', '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': False, 'name': 'A4*D6*A5', 'ngens': 8, '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, 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': 60, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 120, 'number_divisions': 72, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 1209, 'number_subgroup_classes': 1965, 'number_subgroups': 97784, 'old_label': None, 'order': 8640, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 511], [3, 566], [5, 24], [6, 4106], [10, 744], [15, 624], [30, 2064]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [75247205913, 0, 0], 'outer_gens': [[44112104840, 30, 1528450560, 766080, 1078116480, 5015, 998686080, 649595574], [680, 30, 1528450560, 766080, 1078116480, 50339126223, 998686080, 2039788854], [680, 30, 2079342720, 997920000, 124104960, 50339126223, 998686080, 1528087734]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [3, 4], [4, 6], [5, 4], [6, 6], [8, 6], [10, 6], [12, 10], [15, 4], [16, 2], [18, 4], [20, 2], [24, 4], [30, 2], [36, 2]], 'representations': {'Perm': {'d': 14, 'gens': [680, 30, 1528450560, 766080, 1078116480, 50339126223, 998686080, 649595574]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_4\\times D_6\\times A_5', 'transitive_degree': 90, 'wreath_data': None, 'wreath_product': False}
