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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '43200.bt', 'ambient_counter': 46, 'ambient_order': 43200, 'ambient_tex': '(C_5\\times A_4):S_6', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 20, 'counter': 278, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '43200.bt.360.j1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '360.j1.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': 360, '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': '120.20', 'subgroup_hash': 20, 'subgroup_order': 120, 'subgroup_tex': 'C_{15}:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '43200.bt', 'aut_centralizer_order': None, 'aut_label': '360.j1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '7200.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['90.g1.a1', '120.i1.a1', '120.j1.a1', '120.k1.a1', '180.l1.a1'], 'contains': ['720.d1.a1', '720.f1.a1', '720.j1.a1', '1080.n1.a1', '1800.h1.a1'], 'core': '2160.a1.a1', 'coset_action_label': None, 'count': 180, 'diagramx': [1706, -1, 6278, -1, 4290, -1, 8285, -1], 'generators': [87657292800, 15240960, 186810624000, 87672534381, 93887947978], 'label': '43200.bt.360.j1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '180.l1.a1', 'old_label': '360.j1.a1', 'projective_image': '43200.bt', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '360.j1.a1', 'subgroup_fusion': None, 'weyl_group': '40.13'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [2, 2, 4, 10], 'aut_gens': [[992, 566032, 685220], [29760, 566032, 685220], [992, 357520, 685220], [29760, 417098, 446894], [4092, 297923, 685220]], 'aut_group': '160.236', 'aut_hash': 236, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 160, 'aut_permdeg': 11, 'aut_perms': [1, 126, 1169280, 4359888], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 10, 1, 1], [3, 1, 2, 1], [4, 10, 1, 1], [5, 2, 2, 1], [6, 1, 2, 1], [6, 2, 2, 1], [6, 10, 2, 1], [10, 2, 2, 1], [10, 2, 4, 1], [12, 10, 2, 1], [15, 2, 4, 1], [30, 2, 4, 1], [30, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3\\times F_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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 10, 1], [3, 1, 2], [4, 10, 1], [5, 2, 2], [6, 1, 2], [6, 2, 2], [6, 10, 2], [10, 2, 6], [12, 10, 2], [15, 2, 4], [30, 2, 12]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '20.4', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 20, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['40.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 10, 1, 1], [3, 1, 2, 1], [4, 10, 1, 1], [5, 2, 2, 1], [6, 1, 2, 1], [6, 2, 2, 1], [6, 10, 2, 1], [10, 2, 2, 1], [10, 2, 4, 1], [12, 10, 2, 1], [15, 2, 4, 1], [30, 2, 4, 1], [30, 2, 8, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 24, 'exponent': 60, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[2, 0, 8]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '60.10', 'hash': 20, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 10, 5], 'inner_gens': [[992, 89392, 804380], [22940, 566032, 685220], [15438, 566032, 685220]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'D_{10}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 12], [2, 27]], 'label': '120.20', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 10, 'linQ_dim': 8, 'linQ_dim_count': 10, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C15:D4', 'ngens': 3, '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': 16, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 39, 'number_divisions': 16, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 32, 'number_subgroups': 80, 'old_label': None, 'order': 120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 13], [3, 2], [4, 10], [5, 4], [6, 26], [10, 12], [12, 20], [15, 8], [30, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [29792, 29792], 'outer_gens': [[992, 536259, 804380], [992, 417098, 446894]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [4, 3], [8, 3], [16, 1]], 'representations': {'PC': {'code': 10473736188964871, 'gens': [1, 2, 3], 'pres': [5, -2, -2, 2, -3, -5, 1172, 42, 643, 78, 2404]}, 'GLFp': {'d': 2, 'p': 31, 'gens': [992, 566032, 685220]}, 'Perm': {'d': 12, 'gens': [40279687, 7620480, 5760, 87091200, 37]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}:D_4', 'transitive_degree': 60, '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': 120, 'aut_gen_orders': [30, 4, 40, 4, 24], 'aut_gens': [[87657292800, 274467916800, 181073576880, 87179097654, 262013875200, 6714051508], [186810624000, 87657292800, 181073576880, 12944695701, 12454041600, 174364611482], [87657292800, 186810624000, 6721263360, 87186643139, 100111334400, 181073577293], [87657292800, 186810624000, 174367554480, 487353196, 267761894400, 262021904393], [87657292800, 274467916800, 12462070320, 87189943944, 93405312000, 267766692928], [274467916800, 186810624000, 181067402880, 6235372421, 93405312000, 100126575406]], 'aut_group': None, 'aut_hash': 3309493389982338751, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 691200, 'aut_permdeg': 900, 'aut_perms': 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'aut_phi_ratio': 60.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 45, 1, 1], [2, 135, 1, 1], [2, 450, 2, 1], [3, 8, 1, 1], [3, 40, 2, 1], [3, 320, 2, 1], [4, 90, 1, 1], [4, 270, 1, 1], [4, 450, 2, 1], [4, 2700, 1, 2], [5, 2, 2, 1], [5, 144, 1, 1], [5, 144, 4, 1], [6, 120, 2, 1], [6, 360, 1, 1], [6, 3600, 2, 1], [10, 6, 2, 1], [10, 90, 2, 1], [10, 270, 2, 1], [10, 432, 1, 1], [10, 432, 4, 1], [12, 720, 1, 1], [12, 3600, 2, 1], [15, 8, 4, 1], [15, 80, 4, 1], [15, 320, 8, 1], [15, 576, 2, 1], [15, 576, 8, 1], [20, 180, 2, 1], [20, 540, 2, 1], [30, 240, 4, 1], [30, 360, 4, 1], [60, 720, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(F_5\\times S_4).A_6.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': None, 'autcentquo_hash': 3309493389982338751, 'autcentquo_nilpotent': False, 'autcentquo_order': 691200, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(F_5\\times S_4).A_6.C_2^2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 45, 1], [2, 135, 1], [2, 450, 2], [3, 8, 1], [3, 40, 2], [3, 320, 2], [4, 90, 1], [4, 270, 1], [4, 450, 2], [4, 2700, 2], [5, 2, 2], [5, 144, 5], [6, 120, 2], [6, 360, 1], [6, 3600, 2], [10, 6, 2], [10, 90, 2], [10, 270, 2], [10, 432, 5], [12, 720, 1], [12, 3600, 2], [15, 8, 4], [15, 80, 4], [15, 320, 8], [15, 576, 10], [20, 180, 2], [20, 540, 2], [30, 240, 4], [30, 360, 4], [60, 720, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '43200.bt', 'commutator_count': 1, 'commutator_label': '21600.bg', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1', '360.118'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 46, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 45, 1, 1], [2, 135, 1, 1], [2, 450, 1, 2], [3, 8, 1, 1], [3, 40, 1, 2], [3, 320, 1, 2], [4, 90, 1, 1], [4, 270, 1, 1], [4, 450, 1, 2], [4, 2700, 1, 2], [5, 2, 2, 1], [5, 144, 1, 1], [5, 144, 4, 1], [6, 120, 1, 2], [6, 360, 1, 1], [6, 3600, 1, 2], [10, 6, 2, 1], [10, 90, 2, 1], [10, 270, 2, 1], [10, 432, 1, 1], [10, 432, 4, 1], [12, 720, 1, 1], [12, 3600, 1, 2], [15, 8, 4, 1], [15, 80, 2, 2], [15, 320, 4, 2], [15, 576, 2, 1], [15, 576, 8, 1], [20, 180, 2, 1], [20, 540, 2, 1], [30, 240, 2, 2], [30, 360, 4, 1], [60, 720, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 477, 'exponent': 60, 'exponents_of_order': [6, 3, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[30, 1, 4], [48, 0, 4], [54, 1, 2], [60, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '43200.bt', 'hash': 3677769386235727755, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 2, 15, 2, 3, 15], 'inner_gens': [[87657292800, 274467916800, 12465013680, 87179097654, 93405312000, 267769923508], [87657292800, 274467916800, 262024847280, 479808054, 181062604800, 174364611508], [274467916800, 186810624000, 181073576880, 6235372134, 93405312000, 100119363508], [87657292800, 186810624000, 6710820480, 87179097654, 100111334400, 181077845978], [274467916800, 186810624000, 12465013680, 268241702454, 262013875200, 174364611508], [186810624000, 87657292800, 262024847280, 180587237145, 93405312000, 6714051508]], 'inner_hash': 3677769386235727755, 'inner_nilpotent': False, 'inner_order': 43200, 'inner_split': True, 'inner_tex': '(C_5\\times A_4):S_6', 'inner_used': [1, 2, 3, 4, 6], 'irrC_degree': 30, 'irrQ_degree': 60, 'irrQ_dim': 60, 'irrR_degree': 30, 'irrep_stats': [[1, 2], [2, 7], [3, 2], [5, 4], [6, 2], [9, 2], [10, 16], [15, 4], [16, 15], [18, 7], [20, 7], [27, 2], [30, 6], [48, 5], [54, 2], [60, 2]], 'label': '43200.bt', 'linC_count': 48, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 8, 'linQ_dim': 12, 'linQ_dim_count': 8, 'linR_count': 48, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': '(C5*A4):S6', 'ngens': 6, '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, 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': 36, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 85, 'number_divisions': 46, 'number_normal_subgroups': 13, 'number_subgroup_autclasses': 557, 'number_subgroup_classes': 865, 'number_subgroups': 222088, 'old_label': None, 'order': 43200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1083], [3, 728], [4, 6660], [5, 724], [6, 7800], [10, 2892], [12, 7920], [15, 8672], [20, 1440], [30, 2400], [60, 2880]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 584, 0], 'outer_gens': [[87657292800, 186810624000, 6716994480, 87179097654, 100111334400, 181070633908], [87657292800, 274467916800, 181073576880, 87179098198, 262013875200, 6714051821], [87657292800, 274467916800, 181077845760, 87179097654, 262013875200, 6716994868]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [3, 2], [4, 1], [5, 4], [8, 1], [9, 2], [10, 4], [12, 1], [15, 4], [16, 1], [18, 1], [20, 3], [27, 2], [30, 2], [32, 1], [36, 1], [40, 3], [48, 1], [60, 2], [64, 1], [72, 1], [80, 1], [108, 1], [120, 1], [128, 1], [192, 1]], 'representations': {'Perm': {'d': 15, 'gens': [87657292800, 274467916800, 181073576880, 87179097654, 262013875200, 6714051508]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_5\\times A_4):S_6', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
