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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '90720.g', 'ambient_counter': 7, 'ambient_order': 90720, 'ambient_tex': 'A_5\\times {}^2G(2,3)', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 60, 'counter': 29, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '90720.g.84.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '84.a1.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': 84, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': False, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '1080.492', 'subgroup_hash': 492, 'subgroup_order': 1080, 'subgroup_tex': 'S_3\\times \\GL(2,4)', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '90720.g', 'aut_centralizer_order': None, 'aut_label': '84.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '30240.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['28.a1.a1'], 'contains': ['168.b1.a1', '252.a1.a1', '252.b1.a1', '420.a1.a1', '504.a1.a1', '840.c1.a1'], 'core': '1512.a1.a1', 'coset_action_label': None, 'count': 84, 'diagramx': [6130, -1, 5672, -1, 6566, -1, 6384, -1], 'generators': [21674863440, 723311280, 60, 22085976251, 22085976240], 'label': '90720.g.84.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '84.a1.a1', 'old_label': '84.a1.a1', 'projective_image': '90720.g', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '84.a1.a1', 'subgroup_fusion': None, 'weyl_group': '360.121'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [12, 2, 30], 'aut_gens': [[14883220, 3669859], [776948, 18920884], [7706201, 3669918], [776992, 18920983]], 'aut_group': '1440.5849', 'aut_hash': 5849, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1440, 'aut_permdeg': 20, 'aut_perms': [1105105406315604838, 2232016647110940464, 1002367888859742119], 'aut_phi_ratio': 5.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 15, 1, 1], [2, 45, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 20, 1, 1], [3, 20, 2, 1], [3, 40, 1, 1], [3, 40, 2, 1], [5, 12, 2, 1], [6, 3, 2, 1], [6, 15, 2, 1], [6, 30, 1, 1], [6, 30, 2, 1], [6, 45, 2, 1], [6, 60, 1, 1], [6, 60, 2, 1], [10, 36, 2, 1], [15, 12, 4, 1], [15, 24, 2, 1], [15, 24, 4, 1], [30, 36, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6\\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': '720.767', 'autcentquo_hash': 767, 'autcentquo_nilpotent': False, 'autcentquo_order': 720, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times S_5', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 15, 1], [2, 45, 1], [3, 1, 2], [3, 2, 3], [3, 20, 3], [3, 40, 3], [5, 12, 2], [6, 3, 2], [6, 15, 2], [6, 30, 3], [6, 45, 2], [6, 60, 3], [10, 36, 2], [15, 12, 4], [15, 24, 6], [30, 36, 4]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '360.121', 'commutator_count': 1, 'commutator_label': '180.19', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '60.5'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 492, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['6.1', 1], ['60.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 15, 1, 1], [2, 45, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 20, 1, 1], [3, 20, 2, 1], [3, 40, 1, 1], [3, 40, 2, 1], [5, 12, 2, 1], [6, 3, 2, 1], [6, 15, 2, 1], [6, 30, 1, 1], [6, 30, 2, 1], [6, 45, 2, 1], [6, 60, 1, 1], [6, 60, 2, 1], [10, 36, 2, 1], [15, 12, 4, 1], [15, 24, 2, 1], [15, 24, 4, 1], [30, 36, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 228, 'exponent': 30, 'exponents_of_order': [3, 3, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 4], [8, 0, 2], [10, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1080.492', 'hash': 492, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [15, 6], 'inner_gens': [[14883220, 12096056], [1492623, 3669859]], 'inner_hash': 121, 'inner_nilpotent': False, 'inner_order': 360, 'inner_split': True, 'inner_tex': 'S_3\\times A_5', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 12, 'irrep_stats': [[1, 6], [2, 3], [3, 12], [4, 6], [5, 6], [6, 6], [8, 3], [10, 3]], 'label': '1080.492', 'linC_count': 32, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 6, 'linQ_dim': 8, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'S3*GL(2,4)', 'ngens': 2, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 45, 'number_divisions': 24, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 115, 'number_subgroup_classes': 121, 'number_subgroups': 1674, 'old_label': None, 'order': 1080, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 63], [3, 188], [5, 24], [6, 396], [10, 72], [15, 192], [30, 144]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [86, 0], 'outer_gens': [[14883191, 3669921], [776980, 3669859]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 3], [5, 2], [6, 2], [8, 3], [10, 3], [12, 3], [16, 1], [20, 1], [24, 1]], 'representations': {'GLFq': {'d': 4, 'q': 4, 'gens': [1087202305, 1081085697, 3230342786, 1081806593, 1109156353]}, 'Perm': {'d': 11, 'gens': [14883220, 3669859]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times \\GL(2,4)', 'transitive_degree': 30, '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': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 1260, 'aut_gen_orders': [6, 18], 'aut_gens': [[13420073584, 6786219630], [21315934873, 52470270779], [2478476249, 32228098563]], 'aut_group': '181440.h', 'aut_hash': 6736178027719690580, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 181440, 'aut_permdeg': 99, 'aut_perms': [719691797114738565767592242484148372258487024694736819333190199441008872748375547436208483233283275000200820422739361201887336511144678487537464509462074018, 657088266987724634297825159011638697541655872395224624786123036817951891548502066086637955725770342382323908921172121647922041421934686068233050641942687537], 'aut_phi_ratio': 8.75, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [2, 63, 1, 1], [2, 945, 1, 1], [3, 20, 1, 1], [3, 56, 1, 1], [3, 84, 1, 2], [3, 1120, 1, 1], [3, 1680, 1, 2], [5, 12, 2, 1], [6, 252, 1, 2], [6, 840, 1, 1], [6, 1260, 1, 3], [6, 3780, 1, 2], [6, 5040, 1, 2], [7, 216, 1, 1], [9, 168, 1, 3], [9, 3360, 1, 3], [10, 756, 2, 1], [14, 3240, 1, 1], [15, 672, 2, 1], [15, 1008, 2, 2], [18, 2520, 1, 3], [21, 4320, 1, 1], [30, 3024, 2, 2], [35, 2592, 2, 1], [45, 2016, 2, 3]], 'aut_supersolvable': False, 'aut_tex': 'S_5\\times {}^2G(2,3)', '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': 1260, 'autcentquo_group': '181440.h', 'autcentquo_hash': 6736178027719690580, 'autcentquo_nilpotent': False, 'autcentquo_order': 181440, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_5\\times {}^2G(2,3)', 'cc_stats': [[1, 1, 1], [2, 15, 1], [2, 63, 1], [2, 945, 1], [3, 20, 1], [3, 56, 1], [3, 84, 2], [3, 1120, 1], [3, 1680, 2], [5, 12, 2], [6, 252, 2], [6, 840, 1], [6, 1260, 3], [6, 3780, 2], [6, 5040, 2], [7, 216, 1], [9, 168, 3], [9, 3360, 3], [10, 756, 2], [14, 3240, 1], [15, 672, 2], [15, 1008, 4], [18, 2520, 3], [21, 4320, 1], [30, 3024, 4], [35, 2592, 2], [45, 2016, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '90720.g', 'commutator_count': 1, 'commutator_label': '30240.h', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '60.5', '504.156'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['1512.779', 1], ['60.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [2, 63, 1, 1], [2, 945, 1, 1], [3, 20, 1, 1], [3, 56, 1, 1], [3, 84, 2, 1], [3, 1120, 1, 1], [3, 1680, 2, 1], [5, 12, 2, 1], [6, 252, 2, 1], [6, 840, 1, 1], [6, 1260, 1, 1], [6, 1260, 2, 1], [6, 3780, 2, 1], [6, 5040, 2, 1], [7, 216, 1, 1], [9, 168, 1, 1], [9, 168, 2, 1], [9, 3360, 1, 1], [9, 3360, 2, 1], [10, 756, 2, 1], [14, 3240, 1, 1], [15, 672, 2, 1], [15, 1008, 4, 1], [18, 2520, 1, 1], [18, 2520, 2, 1], [21, 4320, 1, 1], [30, 3024, 4, 1], [35, 2592, 2, 1], [45, 2016, 2, 1], [45, 2016, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 21584, 'exponent': 630, 'exponents_of_order': [5, 4, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3, 5, 7], 'faithful_reps': [[21, 0, 4], [21, 1, 2], [24, 0, 4], [24, 1, 2], [28, 0, 2], [28, 1, 1], [32, 0, 2], [32, 1, 1], [35, 0, 2], [35, 1, 1], [40, 0, 2], [40, 1, 1], [63, 1, 2], [81, 1, 2], [84, 1, 1], [105, 1, 1], [108, 1, 1], [135, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '90720.g', 'hash': 6030049094626866441, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 630, 'inner_gen_orders': [30, 6], 'inner_gens': [[13420073584, 40836021123], [20125376020, 6786219630]], 'inner_hash': 6030049094626866441, 'inner_nilpotent': False, 'inner_order': 90720, 'inner_split': True, 'inner_tex': 'A_5\\times {}^2G(2,3)', 'inner_used': [1, 2], 'irrC_degree': 21, 'irrQ_degree': 28, 'irrQ_dim': 28, 'irrR_degree': 21, 'irrep_stats': [[1, 3], [3, 6], [4, 3], [5, 3], [7, 3], [8, 3], [21, 7], [24, 6], [27, 1], [28, 3], [32, 3], [35, 3], [40, 3], [63, 2], [81, 2], [84, 1], [105, 1], [108, 1], [135, 1]], 'label': '90720.g', 'linC_count': 18, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 1, 'linQ_dim': 11, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'A5*2G(2,3)', 'ngens': 2, '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, 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': 44, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 55, 'number_divisions': 32, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 313, 'number_subgroup_classes': 331, 'number_subgroups': 172782, 'old_label': None, 'order': 90720, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1023], [3, 4724], [5, 24], [6, 22764], [7, 216], [9, 10584], [10, 1512], [14, 3240], [15, 5376], [18, 7560], [21, 4320], [30, 12096], [35, 5184], [45, 12096]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [82], 'outer_gens': [[50785822169, 13551074019]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 2], [10, 1], [12, 1], [14, 1], [16, 1], [21, 1], [27, 1], [28, 1], [32, 1], [35, 1], [40, 1], [42, 1], [48, 1], [56, 1], [64, 1], [70, 1], [80, 1], [84, 2], [96, 1], [105, 1], [108, 1], [126, 1], [135, 1], [162, 1]], 'representations': {'Perm': {'d': 14, 'gens': [13420073584, 6786219630]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_5\\times {}^2G(2,3)', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
