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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.4469', 'ambient_counter': 4469, 'ambient_order': 256, 'ambient_tex': 'C_4^2:(C_2\\times C_8)', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 32, 'counter': 25, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '256.4469.4.k1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.k1.b1', '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': 4, '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': '64.69', 'subgroup_hash': 69, 'subgroup_order': 64, 'subgroup_tex': 'C_2^3.C_2^3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.4469', 'aut_centralizer_order': 4, 'aut_label': '4.k1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '512.10494213', 'aut_weyl_index': 16, 'centralizer': '32.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.b1.b1', '8.k1.b1', '8.l1.b1', '8.o1.a1', '8.p1.a1', '8.q1.a1', '8.r1.a1'], 'core': '8.b1.b1', 'coset_action_label': None, 'count': 2, 'diagramx': [3789, -1, 4592, -1, 3788, -1, 4582, -1], 'generators': [104, 16, 2], 'label': '256.4469.4.k1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '4.k1.b1', 'projective_image': '64.90', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.k1.b1', 'subgroup_fusion': None, 'weyl_group': '16.14'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[2577, 3969, 1571, 1817], [2577, 2723, 1571, 1817], [565, 3969, 1571, 811], [2577, 1957, 1571, 1817], [2609, 3969, 1571, 1817], [2577, 4001, 1571, 1817], [2577, 3969, 1571, 1849], [3603, 3969, 1571, 1817], [2577, 3969, 1571, 811], [2577, 3969, 1571, 3869]], 'aut_group': '512.10494213', 'aut_hash': 1718285292446712970, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 18, 'aut_perms': [1, 6, 120, 5040, 362880, 39916800, 6227020800, 1307674368000, 355687428096000], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 4, 2, 1], [4, 2, 4, 3], [4, 4, 2, 3]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '512.10494213', 'autcent_hash': 1718285292446712970, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^9', '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, 7], [2, 4, 2], [4, 2, 12], [4, 4, 6]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, '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': 69, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 4, 1, 2], [4, 2, 2, 6], [4, 4, 1, 2], [4, 4, 2, 2]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 168, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '8.5', 'hash': 69, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 2], 'inner_gens': [[2577, 1957, 1571, 1817], [565, 3969, 1571, 811], [2577, 3969, 1571, 1817], [2577, 2723, 1571, 1817]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': True, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12]], 'label': '64.69', 'linC_count': 384, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 16, 'linQ_dim': 6, 'linQ_dim_count': 16, 'linR_count': 16, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.C2^3', 'ngens': 3, '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': 15, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 28, 'number_divisions': 20, 'number_normal_subgroups': 45, 'number_subgroup_autclasses': 75, 'number_subgroup_classes': 95, 'number_subgroups': 161, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15], [4, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2, 2], 'outer_gen_pows': [513, 513, 513, 513, 513, 513], 'outer_gens': [[2577, 3969, 1571, 3869], [2577, 3969, 1571, 811], [3603, 3969, 1571, 1817], [2577, 3969, 1571, 1849], [2577, 4001, 1571, 1817], [2609, 3969, 1571, 1817]], 'outer_group': '64.267', 'outer_hash': 267, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 12, 'outer_perms': [39916800, 362880, 5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 4]], 'representations': {'PC': {'code': 1103890514001, 'gens': [1, 3, 4, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 12, 650, 196, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101736067516548, 91655872443234552, 25038659807091718]}, 'GLFq': {'d': 3, 'q': 8, 'gens': [16849409, 16880641, 101521990, 2036992, 84252549, 16979969]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [1817, 2565, 1941, 545, 1539, 1693]}, 'Perm': {'d': 12, 'gens': [40285562, 5890, 7620493, 23, 87091223, 11520]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3.C_2^3', 'transitive_degree': 32, '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': '32.36', '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, 4, 4, 4, 4, 2], 'aut_gens': [[1, 8, 16, 64], [121, 104, 80, 64], [25, 76, 80, 64], [5, 8, 212, 224], [169, 136, 180, 224], [109, 172, 244, 192], [11, 12, 16, 192], [3, 136, 144, 224]], 'aut_group': None, 'aut_hash': 2254421811164287685, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 128, 'aut_perms': [35202734196200699092594787359132377448661708659749649876186383658574040952016874405852882889094957802565506161078081949171908907315794314246634263204590749656295138765506187372002551478228343943824525428969905480413, 382793020662324565381184825273944802313904857644432399359808718167208757566933346499994480976411851243585792702840962855949288549562700132713714393307533546013057012008027682266593377303400357635211155689482811345885, 145399587463320964214133675918424669528606107221868282188519717774731083411978347745526766679915514426746189708509685741904865289289252100589859986961927660160045835245142959103985940167483515539960488346569680269088, 296162914653012696704345948380055623353980964455115528624172891407437551844620916524927022475658274720026490809401567628422070060923324178132857492229197327332640489896792507828898885163504692935247143654953402692734, 201356115659348309165867653036273180403850182563014193907320195372740677256213917600255234610941572305400151299728737108980114936692256136029222614477493914167837815642405516651012459491864860643247767020044928018866, 262382377451776181736177232832635177558731887277138304820696575745867743525526536443899310622722218245343448418923793621373738252301604027486220725474875411863509990866174115004790809574962394867976208766059176565464, 373787656127067895825884750751225079023580588350935771924352876168728447465187351843042691616710039550887794866820376486976679777610051592908337703451488285969608214631231462198728917700125826159664478683207489695153], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 4, 1], [4, 2, 4, 1], [4, 4, 1, 2], [4, 4, 2, 1], [4, 4, 4, 1], [4, 8, 2, 4], [8, 8, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2.C_2^6.C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '128.1755', 'autcentquo_hash': 1755, 'autcentquo_nilpotent': True, 'autcentquo_order': 128, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^4:D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 4], [4, 2, 4], [4, 4, 8], [4, 8, 8], [8, 8, 16]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '64.90', 'commutator_count': 1, 'commutator_label': '8.2', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4469, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 4], [4, 2, 2, 2], [4, 4, 1, 2], [4, 4, 2, 3], [4, 8, 1, 4], [4, 8, 2, 2], [8, 8, 4, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 672, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '32.23', 'frattini_quotient': '8.5', 'hash': 4469, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 2, 4, 2], 'inner_gens': [[1, 8, 112, 96], [1, 8, 176, 192], [225, 168, 16, 64], [33, 136, 16, 64]], 'inner_hash': 90, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': False, 'inner_tex': 'C_2^4:C_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8], [4, 4], [8, 2]], 'label': '256.4469', 'linC_count': 32, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 12, 'linQ_dim': 12, 'linQ_dim_count': 6, 'linR_count': 8, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2:(C2*C8)', 'ngens': 3, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 17, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 46, 'number_divisions': 27, 'number_normal_subgroups': 65, 'number_subgroup_autclasses': 135, 'number_subgroup_classes': 230, 'number_subgroups': 663, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 23], [4, 104], [8, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [50, 0, 0, 0, 0, 0, 0], 'outer_gens': [[21, 108, 80, 224], [11, 8, 16, 192], [7, 8, 20, 224], [1, 140, 16, 64], [37, 8, 148, 64], [5, 8, 16, 64], [5, 136, 16, 64]], 'outer_group': '128.2216', 'outer_hash': 2216, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [139361783, 167880, 167450525, 303182776, 90842416, 167450520, 16], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4^2:C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 8], [8, 3]], 'representations': {'PC': {'code': 42350315019969407431943062766820435, 'gens': [1, 4, 5, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 16, 41, 4484, 972, 908, 116, 7685, 5382, 5390, 1374, 166]}, 'Perm': {'d': 24, 'gens': [30514592371074684371731, 51972307879040458560000, 85885156128067346144530, 112938477265386505514880, 139968018576646777821655, 163455923581589024830080, 24893, 183273319277893773290880]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2:(C_2\\times C_8)', 'transitive_degree': 64, 'wreath_data': None, 'wreath_product': False}
