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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1728.47309', 'ambient_counter': 47309, 'ambient_order': 1728, 'ambient_tex': 'C_{12}.D_6^2', 'central': False, 'central_factor': False, 'centralizer_order': 432, 'characteristic': False, 'core_order': 4, 'counter': 607, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.47309.144.i1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '144.i1', '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': '12.5', 'subgroup_hash': 5, 'subgroup_order': 12, 'subgroup_tex': 'C_2\\times C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.47309', 'aut_centralizer_order': None, 'aut_label': '144.i1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '4.c1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['48.b1', '48.m1', '48.o1', '72.i1', '72.l1', '72.t1', '72.bf1', '72.bg1', '72.bh1', '72.bi1'], 'contains': ['288.f1', '288.i1', '432.a1'], 'core': '432.a1', 'coset_action_label': None, 'count': 2, 'diagramx': None, 'generators': [144, 864, 104], 'label': '1728.47309.144.i1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '48.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1', 'old_label': '144.i1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '144.i1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [6, 9], [6, 11]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 31], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [6, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.4', 'autcent_hash': 4, 'autcent_nilpotent': False, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_6', '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, 3], [3, 1, 2], [6, 1, 6]], 'center_label': '12.5', 'center_order': 12, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [6, 1, 2, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '12.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 12]], 'label': '12.5', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 6, 'linQ_dim': 3, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C6', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 12, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 9], [6, 11]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4]], 'representations': {'PC': {'code': 273, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16507]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 2064]}, 'Perm': {'d': 7, 'gens': [720, 24, 4]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_6', 'transitive_degree': 12, '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': 12, 'aut_gen_orders': [4, 6, 6, 12, 2, 12, 6, 12, 12], 'aut_gens': [[1, 2, 4, 24, 288], [1009, 1106, 308, 120, 1584], [953, 1026, 1444, 264, 1440], [1025, 1122, 580, 984, 288], [1059, 1058, 924, 944, 288], [1, 242, 292, 264, 1440], [1025, 162, 1444, 168, 1584], [161, 66, 1156, 1128, 1440], [945, 154, 436, 1032, 288], [1089, 250, 308, 120, 288]], 'aut_group': None, 'aut_hash': 6597356278214586319, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 442368, 'aut_permdeg': 60, 'aut_perms': [3882800480156121397150857467263332144896159365198594723645618694650772942479691204, 2491505763578137692689647008922117660860704751038917739936326641097273035186151375, 4014675711136715440309750800006281205351530380302966886846513248643538847259914616, 5657413109753308249097442702209598930828013742936135424949581284516016481910350209, 138644518034313818094889420947371562813213875995485565652765566603915173677794151, 4016580093305093370891261019871710784950890542492257097243367795883796308670806707, 3319111090340506444233490645260053535244248990759127029617080908905552911854735389, 6503488018405824514182610159211469742018397485612330637247523139036180339974911960, 1255064274063668951937350836244633451618934271414022849721492751659488487892336193], 'aut_phi_ratio': 768.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 6, 2, 1], [2, 9, 4, 1], [2, 18, 8, 1], [2, 54, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [4, 2, 2, 1], [4, 3, 4, 1], [4, 6, 8, 1], [4, 18, 2, 1], [4, 27, 4, 1], [6, 2, 1, 1], [6, 2, 2, 2], [6, 2, 4, 1], [6, 4, 1, 1], [6, 4, 2, 2], [6, 4, 4, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 12, 4, 2], [6, 18, 4, 1], [6, 36, 8, 1], [12, 4, 2, 1], [12, 4, 4, 2], [12, 6, 8, 1], [12, 8, 4, 2], [12, 12, 4, 1], [12, 12, 8, 2], [12, 24, 8, 1], [12, 36, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^3.C_2^6.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '512.6249567', 'autcent_hash': 6452065007057432321, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '864.4686', 'autcentquo_hash': 4686, 'autcentquo_nilpotent': False, 'autcentquo_order': 864, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^3:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 6, 2], [2, 9, 4], [2, 18, 8], [2, 54, 2], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 2, 2], [4, 3, 4], [4, 6, 8], [4, 18, 2], [4, 27, 4], [6, 2, 9], [6, 4, 9], [6, 8, 3], [6, 12, 8], [6, 18, 4], [6, 36, 8], [12, 4, 10], [12, 6, 8], [12, 8, 8], [12, 12, 20], [12, 24, 8], [12, 36, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '432.759', 'commutator_count': 1, 'commutator_label': '54.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 47309, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['864.4358', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 6, 1, 2], [2, 9, 1, 4], [2, 18, 1, 8], [2, 54, 1, 2], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 2, 1, 2], [4, 3, 2, 2], [4, 6, 1, 8], [4, 18, 1, 2], [4, 27, 2, 2], [6, 2, 1, 9], [6, 4, 1, 9], [6, 8, 1, 3], [6, 12, 1, 8], [6, 18, 1, 4], [6, 36, 1, 8], [12, 4, 1, 6], [12, 4, 2, 2], [12, 6, 2, 4], [12, 8, 1, 4], [12, 8, 2, 2], [12, 12, 1, 16], [12, 12, 2, 2], [12, 24, 1, 8], [12, 36, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 9999360000, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4704', 'hash': 47309, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 6, 6, 3], 'inner_gens': [[1, 2, 164, 168, 288], [1, 2, 20, 120, 288], [153, 10, 4, 24, 1440], [145, 194, 4, 24, 288], [1, 2, 580, 24, 288]], 'inner_hash': 759, 'inner_nilpotent': False, 'inner_order': 432, 'inner_split': False, 'inner_tex': 'C_2\\times S_3^3', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 56], [4, 44], [8, 12]], 'label': '1728.47309', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C12.D6^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 42, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 144, 'number_divisions': 130, 'number_normal_subgroups': 668, 'number_subgroup_autclasses': 744, 'number_subgroup_classes': 4178, 'number_subgroups': 31092, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 303], [3, 26], [4, 208], [6, 534], [12, 656]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 2, 4, 2, 4, 4, 2], 'outer_gen_pows': [0, 0, 0, 218, 0, 0, 0], 'outer_gens': [[1, 866, 164, 1032, 1584], [1, 2, 20, 984, 1440], [1081, 2, 164, 1032, 1584], [75, 2, 108, 224, 1440], [73, 866, 4, 120, 432], [3, 146, 204, 952, 1584], [1009, 1010, 1012, 120, 288]], 'outer_group': '5832.jm', 'outer_hash': 2233397451516965271, 'outer_nilpotent': True, 'outer_order': 1024, 'outer_permdeg': 128, 'outer_perms': [355325380253272625766399487646943165983490149071273741974351310662466166004099233797652641043196609917617793427305680775177480703497542235737705932583991739249007833666239796841260599642509767449259129771130890707743, 12266607246179434999527786088310700616923004488478819869281818218806465993026864486178358064217853048759733124963847190726663614545231328457501002726181202895941956862413405143195337556555532831718711379250932870902, 58126346006888136518481837881180785045591305177774183314137250007818012757324062791018949627678665685534711179732534037835019589049856113083460716257266042396510073810600849732011278682042930613515562366566016963556, 21376322088233837366079369574171611808308217469833858671748884854486161462932669673718241474392626341993412671794587791833573569634161345079791353260308422380145152879140597959036800107178411468949327895605454972796, 330988130595993714026924455928382060446242360657552956066418739106982552719419911332082580228819259903898195295628315839355812269769614881172735071033133732857630493606030734211096015508720948448018431621311015564564, 227338039176496628994505585107650771633496125934823453481347387355412427727133697627700861503980715301762857047649805163131551431286329706083151784333655823489918904425874698586671873152709934846171711209116054362046, 182841126912735001688430669683877516739050106534358253294958532284111040590827594840472578027635964666209954140730623618883984599623766330814737662405875302936716756922805721898662147301294137285890800234474489676656], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^5:D_{12}', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 48], [4, 32], [8, 16], [16, 2]], 'representations': {'PC': {'code': 19085579094396486051505698202506083568082714347036360268671525, 'gens': [1, 2, 3, 5, 8], 'pres': [9, -2, -2, -2, -3, -2, -2, -3, -2, -3, 1296, 4430, 281, 74, 579, 300, 7564, 2713, 130, 6494, 158, 6063, 25945, 214, 23354]}, 'Perm': {'d': 19, 'gens': [1134265, 483094080, 356995106556480, 1307682729600, 1307686879680, 8361600, 6706022400, 3, 6758061133824000]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}.D_6^2', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
