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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '10368.qo', 'ambient_counter': 431, 'ambient_order': 10368, 'ambient_tex': '\\SOPlus(4,2)^2.C_2', 'central': False, 'central_factor': False, 'centralizer_order': 72, 'characteristic': False, 'core_order': 9, 'counter': 666, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '10368.qo.144.ci1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '144.ci1', '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': 144, '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': '72.14', 'subgroup_hash': 14, 'subgroup_order': 72, 'subgroup_tex': 'C_3\\times C_{24}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '10368.qo', 'aut_centralizer_order': None, 'aut_label': '144.ci1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '144.ci1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['16.l1', '72.v1', '72.bx1', '72.ch1', '72.cl1', '72.cr1'], 'contains': ['288.bj1', '432.bw1'], 'core': '1152.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [7841, -1, 6794, -1], 'generators': [15, 21675270960, 8237376000, 373, 39330028080], 'label': '10368.qo.144.ci1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '16.l1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.b1', 'old_label': '144.ci1', 'projective_image': '10368.qo', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '144.ci1', 'subgroup_fusion': None, 'weyl_group': '16.11'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '72.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [4, 6, 2, 4, 2, 2, 2], 'aut_gens': [[1, 3], [24, 28], [48, 35], [2, 4], [26, 5], [2, 33], [1, 39], [1, 57]], 'aut_group': '192.1475', 'aut_hash': 1475, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 12, 'aut_perms': [87465720, 174182407, 174988943, 262903080, 40284847, 16, 7], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 8, 1], [4, 1, 2, 1], [6, 1, 8, 1], [8, 1, 4, 1], [12, 1, 16, 1], [24, 1, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times \\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '192.1475', 'autcent_hash': 1475, 'autcent_nilpotent': False, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^2\\times \\GL(2,3)', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 8], [4, 1, 2], [6, 1, 8], [8, 1, 4], [12, 1, 16], [24, 1, 32]], 'center_label': '72.14', 'center_order': 72, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 2], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [4, 1, 2, 1], [6, 1, 2, 4], [8, 1, 4, 1], [12, 1, 4, 4], [24, 1, 8, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 12, 'exponent': 24, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '18.5', 'hash': 14, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 72]], 'label': '72.14', 'linC_count': 1152, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 24, 'linQ_dim': 8, 'linQ_dim_count': 24, 'linR_count': 288, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C24', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 72, 'number_divisions': 20, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 24, 'number_subgroups': 24, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 2], [6, 8], [8, 4], [12, 16], [24, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [4, 6, 2, 4, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[24, 28], [48, 35], [2, 4], [26, 5], [2, 33], [1, 39], [1, 57]], 'outer_group': '192.1475', 'outer_hash': 1475, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 12, 'outer_perms': [87465720, 174182407, 174988943, 262903080, 40284847, 16, 7], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^2\\times \\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [8, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 5], [8, 4]], 'representations': {'PC': {'code': 2834301393, 'gens': [1, 2], 'pres': [5, -3, -2, -2, -2, -3, 26, 42, 58]}, 'GLFp': {'d': 2, 'p': 73, 'gens': [19061836, 3890222]}, 'Perm': {'d': 14, 'gens': [46637337600, 240, 4, 19640234880, 6267305520]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 24], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_{24}', 'transitive_degree': 72, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [8, 8, 12, 8, 4], 'aut_gens': [[55, 107935072560, 208417633200, 186, 20, 25032193516, 8237376000, 14936145840, 13579781094], [403, 208417633200, 350350479360, 15, 378, 387519597381, 289397203200, 262660970880, 697625994556], [67, 208417633200, 555770033280, 181, 15, 619350203338, 268931537280, 270726986880, 182111695974], [60, 655840649520, 450751835520, 373, 366, 264017336339, 120233433600, 94933036800, 623823419534], [60, 208417633200, 555770033280, 174, 378, 94455209574, 697625994240, 374151461040, 289397203859], [403, 299510305920, 655840649520, 15, 186, 46463558421, 13579781040, 14936145840, 8237376719]], 'aut_group': None, 'aut_hash': 2723700116397070647, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20736, 'aut_permdeg': 42, 'aut_perms': [130464360219826856029674814068861990777124882991480, 344278660541932616747885775268561816191527414885975, 347622154334153664000197527635309835759639941002759, 310139888288768074471352824155073463450625365151793, 495964547811237692664051156303754274909148729688058], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 6, 2, 1], [2, 9, 1, 2], [2, 12, 1, 1], [2, 54, 2, 1], [2, 72, 2, 1], [2, 81, 1, 1], [2, 108, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [3, 32, 2, 1], [4, 18, 1, 2], [4, 36, 1, 1], [4, 108, 2, 1], [4, 162, 1, 2], [4, 216, 1, 1], [4, 216, 2, 1], [4, 324, 1, 2], [4, 648, 1, 1], [6, 12, 2, 1], [6, 24, 1, 1], [6, 36, 2, 1], [6, 48, 2, 2], [6, 72, 1, 1], [6, 96, 2, 2], [6, 108, 2, 1], [6, 144, 2, 2], [6, 216, 1, 1], [6, 288, 2, 1], [8, 18, 1, 2], [8, 108, 2, 2], [8, 162, 1, 2], [8, 324, 1, 2], [12, 72, 2, 1], [12, 144, 1, 1], [12, 144, 2, 1], [12, 216, 2, 1], [12, 432, 1, 1], [12, 432, 2, 1], [24, 72, 2, 2], [24, 216, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_4^2.C_2^4', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 2723700116397070647, 'autcentquo_nilpotent': False, 'autcentquo_order': 20736, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_4^2.C_2^4', 'cc_stats': [[1, 1, 1], [2, 6, 2], [2, 9, 2], [2, 12, 1], [2, 54, 2], [2, 72, 2], [2, 81, 1], [2, 108, 1], [3, 4, 2], [3, 8, 1], [3, 32, 2], [4, 18, 2], [4, 36, 1], [4, 108, 2], [4, 162, 2], [4, 216, 3], [4, 324, 2], [4, 648, 1], [6, 12, 2], [6, 24, 1], [6, 36, 2], [6, 48, 4], [6, 72, 1], [6, 96, 4], [6, 108, 2], [6, 144, 4], [6, 216, 1], [6, 288, 2], [8, 18, 2], [8, 108, 4], [8, 162, 2], [8, 324, 2], [12, 72, 2], [12, 144, 3], [12, 216, 2], [12, 432, 3], [24, 72, 4], [24, 216, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '10368.qo', 'commutator_count': 1, 'commutator_label': '648.715', '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', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 431, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['144.182', 1], ['72.40', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 6, 1, 2], [2, 9, 1, 2], [2, 12, 1, 1], [2, 54, 1, 2], [2, 72, 1, 2], [2, 81, 1, 1], [2, 108, 1, 1], [3, 4, 1, 2], [3, 8, 1, 1], [3, 32, 1, 2], [4, 18, 1, 2], [4, 36, 1, 1], [4, 108, 1, 2], [4, 162, 1, 2], [4, 216, 1, 3], [4, 324, 1, 2], [4, 648, 1, 1], [6, 12, 1, 2], [6, 24, 1, 1], [6, 36, 1, 2], [6, 48, 1, 4], [6, 72, 1, 1], [6, 96, 1, 4], [6, 108, 1, 2], [6, 144, 1, 4], [6, 216, 1, 1], [6, 288, 1, 2], [8, 18, 2, 1], [8, 108, 2, 2], [8, 162, 2, 1], [8, 324, 2, 1], [12, 72, 1, 2], [12, 144, 1, 3], [12, 216, 1, 2], [12, 432, 1, 3], [24, 72, 2, 2], [24, 216, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 170951639040, 'exponent': 24, 'exponents_of_order': [7, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[32, 1, 8]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '10368.qo', 'hash': 4072227654749320783, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 3, 3, 3, 3, 8, 8, 4, 8], 'inner_gens': [[55, 107935072560, 208417633200, 373, 15, 25032193516, 8237376000, 14936145840, 13579781094], [55, 107935072560, 208417633200, 186, 20, 619350203116, 268931537280, 181633132800, 182111695974], [55, 107935072560, 208417633200, 186, 20, 455730186556, 374630140080, 287601637680, 548951437494], [396, 107935072560, 208417633200, 186, 20, 25032193798, 8237376000, 14936145840, 13579781430], [60, 107935072560, 208417633200, 186, 20, 25032193859, 8237376000, 14936145840, 13579781094], [55, 450751835520, 107935072560, 373, 366, 25032193516, 8237376000, 43789823280, 13579781041], [55, 208417633200, 555770033280, 186, 20, 25032193516, 8237376000, 7320463920, 13579781094], [55, 299510305920, 655840649520, 186, 20, 46463558716, 13579781040, 14936145840, 8237376054], [55, 299510305920, 350350479360, 378, 20, 25032193541, 8237376000, 43789823280, 13579781094]], 'inner_hash': 4072227654749320783, 'inner_nilpotent': False, 'inner_order': 10368, 'inner_split': False, 'inner_tex': '\\SOPlus(4,2)^2.C_2', 'inner_used': [1, 2, 3, 4, 5, 6, 7, 8, 9], 'irrC_degree': 32, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 32, 'irrep_stats': [[1, 16], [2, 16], [4, 19], [8, 20], [16, 2], [32, 8]], 'label': '10368.qo', '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': None, 'name': 'SO+(4,2)^2.C2', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 54, 'number_characteristic_subgroups': 64, 'number_conjugacy_classes': 81, 'number_divisions': 72, 'number_normal_subgroups': 118, 'number_subgroup_autclasses': 1318, 'number_subgroup_classes': 2197, 'number_subgroups': 105240, 'old_label': None, 'order': 10368, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 483], [3, 80], [4, 2556], [6, 2352], [8, 1440], [12, 2304], [24, 1152]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [55], 'outer_gens': [[55, 107935072560, 208417633200, 366, 181, 25032193254, 8237376000, 14936145840, 13579781356]], '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': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 21], [8, 13], [16, 6], [32, 8]], 'representations': {'PC': {'code': '2582467937227575674226118623295126587000088774182148228047009580157038234771536636310276251827157607484859552583325213669433648171277254898719591430665565288487', 'gens': [1, 2, 5, 7, 10, 11], 'pres': [11, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 88, 133, 56, 398, 90, 22004, 11015, 6406, 587, 158, 52805, 26416, 13755, 566, 41894, 13569, 3108, 4659, 3284, 226, 87303, 26770, 4957, 10600, 260, 82376, 3187, 11118, 9545, 6675, 1406, 557589, 69750, 5884, 4443]}, 'Perm': {'d': 15, 'gens': [55, 107935072560, 208417633200, 186, 20, 25032193516, 8237376000, 14936145840, 13579781094]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SOPlus(4,2)^2.C_2', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
