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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1024.ddn', 'ambient_counter': 2120, 'ambient_order': 1024, 'ambient_tex': '(C_2\\times D_4^2):D_4', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 16, 'counter': 565, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1024.ddn.16.FX', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '16.fr1', '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': 16, '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': '64.202', 'subgroup_hash': 202, 'subgroup_order': 64, 'subgroup_tex': 'C_2^3:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1024.ddn', 'aut_centralizer_order': None, 'aut_label': '16.FX', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '64.DT', 'complements': None, 'conjugacy_class_count': 8, 'contained_in': ['8.BY', '8.CG', '8.CM', '8.CO', '8.FB', '8.FH', '8.FI'], 'contains': ['32.CR', '32.CT', '32.CU', '32.CZ', '32.DB', '32.DK', '32.DM', '32.PX', '32.PZ', '32.QB', '32.QE', '32.VM', '32.VR', '32.BBI', '32.BBM'], 'core': '64.F', 'coset_action_label': None, 'count': 16, 'diagramx': [2064, -1, 2065, -1], 'generators': [40284840, 1313901388800, 302814840, 2789792426303, 18132997461703, 1313941673647], 'label': '1024.ddn.16.FX', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.BB', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.C', 'old_label': '16.fr1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.FX', 'subgroup_fusion': None, 'weyl_group': None}
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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': 4, 'aut_exponent': 24, 'aut_gen_orders': [8, 4], 'aut_gens': [[4001, 643, 2565, 1571, 3993], [517, 2727, 3623, 545, 687], [1957, 1575, 3623, 1571, 1561]], 'aut_group': '12288.dp', 'aut_hash': 1630168457023225010, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 24, 'aut_perms': [187522706853152502407387, 128367454260810836992011], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 1, 4, 1], [2, 2, 12, 1], [2, 4, 2, 1], [4, 4, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^9.S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 8092891664598236742, 'autcent_nilpotent': True, 'autcent_order': 2048, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8.C_2^3', '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, 2, 12], [2, 4, 2], [4, 4, 6]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 202, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['32.27', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 12], [2, 4, 1, 2], [4, 4, 1, 6]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 420, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '16.14', 'hash': 202, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 1, 2], 'inner_gens': [[4001, 643, 2565, 1571, 2731], [4001, 643, 2565, 1571, 2699], [4001, 643, 2565, 1571, 3993], [4001, 643, 2565, 1571, 3993], [2691, 1921, 2565, 1571, 3993]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12]], 'label': '64.202', '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': 'C2^3:D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 28, 'number_divisions': 28, 'number_normal_subgroups': 105, 'number_subgroup_autclasses': 38, 'number_subgroup_classes': 331, 'number_subgroups': 569, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 39], [4, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [513, 513, 513, 513, 513, 513, 513, 513, 513, 513], 'outer_gens': [[1575, 643, 2565, 545, 3993], [2695, 3619, 2565, 1539, 1593], [2723, 675, 2565, 1571, 1981], [4001, 643, 3623, 1571, 3993], [2723, 1921, 3591, 1571, 3993], [1957, 2695, 2565, 1571, 1981], [1957, 643, 2565, 1571, 3993], [4001, 1921, 2565, 1571, 3993], [4001, 1953, 2565, 1571, 3993], [2723, 643, 2565, 1571, 3993]], 'outer_group': '1536.408632814', 'outer_hash': 2972356385187470528, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 14, 'outer_perms': [1683987865, 7878390024, 21317548327, 14390022966, 21797289414, 243538566, 85686, 2438997120, 21316418640, 2440489806], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12]], 'representations': {'PC': {'code': 4564455696, 'gens': [1, 2, 3, 4, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 724, 730, 88]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [705686951426, 141602720900, 706461793378, 706204576418]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [11040833, 31029334, 19896385, 10743875, 6210625, 19208257]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [643, 2565, 523, 545, 1539, 687]}, 'Perm': {'d': 10, 'gens': [367925, 766087, 85800, 806520, 806416, 806400]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3:D_4', 'transitive_degree': 16, '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [4, 4, 8, 4, 8, 8], 'aut_gens': [[5167, 87102727, 302814840, 1313901388800, 5606321846392, 12526762728376], [4103693809943, 127387567, 19614852622078, 127387560, 12520495790657, 15316514841600], [16825370284657, 18126770083062, 1314028776367, 16825147303657, 14008840836601, 5612588778958], [19615028054278, 18126770083062, 1313941673640, 176158200, 12526762728376, 14008840836601], [4103693809943, 4103646986903, 1314164654647, 4103646992063, 9703614810942, 6920263514872], [16825370284657, 20916522950273, 2789705318416, 302814840, 8395940079943, 11212821059896], [18126857180862, 18126730524222, 5160, 16825107018817, 12520495790657, 14002613452807]], 'aut_group': '65536.z', 'aut_hash': 7116534239874182336, 'aut_nilpotency_class': 6, 'aut_nilpotent': True, 'aut_order': 65536, 'aut_permdeg': 64, 'aut_perms': [52591626948307115290803812964962180716499066940416715786308353773404458532969890448490811, 79733326356524296867543537387297126645539955715736747972024602866386158157742232478556858, 22751799738223641498436241376578862879180243805052763503992995566977967652642691723433401, 101169841853168296652318344727987780764917922908601714227056132654911963072060055399938585, 44846508443645655065913432885788396810722743632805870966401208460230441877411542988798238, 61413131909602486310467318809265549046621126430585046183456248624528609677139695243181437], 'aut_phi_ratio': 128.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 2, 2, 1], [2, 4, 2, 1], [2, 4, 4, 2], [2, 4, 8, 1], [2, 8, 2, 2], [2, 8, 4, 3], [2, 16, 2, 1], [4, 4, 2, 1], [4, 4, 4, 2], [4, 8, 1, 1], [4, 8, 2, 2], [4, 8, 4, 3], [4, 8, 8, 2], [4, 16, 2, 1], [4, 32, 2, 1], [4, 32, 8, 1], [8, 32, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9.C_2\\wr D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': None, 'autcentquo_hash': 7822646973264783822, 'autcentquo_nilpotent': True, 'autcentquo_order': 2048, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^5.D_4^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 4, 18], [2, 8, 16], [2, 16, 2], [4, 4, 10], [4, 8, 33], [4, 16, 2], [4, 32, 10], [8, 32, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '512.6536092', 'commutator_count': 1, 'commutator_label': '32.46', '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', '2.1', '2.1', '2.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 2120, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 4, 1, 18], [2, 8, 1, 16], [2, 16, 1, 2], [4, 4, 1, 10], [4, 8, 1, 33], [4, 16, 1, 2], [4, 32, 1, 10], [8, 32, 1, 4]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 5119672320, 'exponent': 8, 'exponents_of_order': [10], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[8, 1, 8]], 'familial': False, 'frattini_label': '32.46', 'frattini_quotient': '32.51', 'hash': 7003003021026862916, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 2, 4, 4], 'inner_gens': [[5167, 87102727, 263265840, 1313901388800, 5606321846392, 11212821054743], [5167, 87102727, 263265840, 1313901388800, 8395940079943, 15316514846767], [40279687, 127387567, 302814840, 1313901388800, 15310288183806, 9703614815857], [5167, 87102727, 302814840, 1313901388800, 6920263514872, 11212861334416], [5167, 2789705323576, 18126554366022, 40279680, 5606321846392, 12526722443536], [1313941668480, 2789745598096, 18126554366022, 5160, 5606362120912, 12526762728376]], 'inner_hash': 3725227854009754059, 'inner_nilpotent': True, 'inner_order': 512, 'inner_split': False, 'inner_tex': 'C_2^6:C_2^3', 'inner_used': [1, 2, 3, 4, 5, 6], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 32], [2, 56], [8, 12]], 'label': '1024.ddn', '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': True, 'name': '(C2*D4^2):D4', 'ngens': 6, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [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': 29, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 100, 'number_divisions': 100, 'number_normal_subgroups': 947, 'number_subgroup_autclasses': 3360, 'number_subgroup_classes': 22344, 'number_subgroups': 79669, 'old_label': None, 'order': 1024, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 239], [4, 656], [8, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 16819007022583, 0, 0, 0, 0, 0], 'outer_gens': [[4103734094783, 1314028776360, 18126594650862, 1314028776367, 6920263509719, 9703654364857], [20916435842633, 16825322735857, 4103646992063, 215717040, 6913996572238, 12520495790657], [5167, 1313988491520, 1314164654647, 1313901388800, 5612549225033, 11219087992462], [5167, 127387567, 1314164654647, 40284847, 5612549225033, 11219087992462], [2789792421136, 87102727, 20916299969273, 127387560, 6920263509719, 9703654369782], [5167, 87102727, 302814840, 40284847, 5606321846392, 12526762728376], [5167, 87102727, 1314164654647, 40284847, 5606321846392, 12526762728376]], 'outer_group': '128.1755', 'outer_hash': 1755, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 10, 'outer_perms': [126745, 1951327, 26334, 1, 2718174, 167280, 1275486], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 56], [8, 12]], 'representations': {'PC': {'code': '31859800216577785331555308478271677616527907748749314695626699018286266766861952945339170', 'gens': [1, 2, 4, 5, 6, 8, 9], 'pres': [10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 121, 51, 20803, 10413, 743, 29604, 2814, 3734, 5765, 2895, 755, 405, 175, 51207, 39057, 7077, 3247, 1657, 69128, 12978, 8678, 4368, 618, 268, 28819]}, 'Perm': {'d': 16, 'gens': [87102727, 1313901388800, 12526762728376, 302814840, 5167, 5606321846392]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times D_4^2):D_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
