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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '15552.fa', 'ambient_counter': 131, 'ambient_order': 15552, 'ambient_tex': 'C_6^4:D_6', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 1, 'counter': 1026, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '15552.fa.216.fh1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '216.fh1', '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': 216, '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': '72.29', 'subgroup_hash': 29, 'subgroup_order': 72, 'subgroup_tex': 'C_6:C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '15552.fa', 'aut_centralizer_order': None, 'aut_label': '216.fh1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1296.bf1', 'complements': None, 'conjugacy_class_count': 6, 'contained_in': ['72.cu1', '72.cy1', '72.da1', '108.g1', '108.v1', '108.em1'], 'contains': ['432.da1', '432.jr1', '648.cv1', '648.cz1'], 'core': '15552.a1', 'coset_action_label': None, 'count': 324, 'diagramx': None, 'generators': [79833774, 648148806726220800, 852272846361171052, 757343707607102710, 318533], 'label': '15552.fa.216.fh1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '54.bu1', 'old_label': '216.fh1', 'projective_image': '15552.fa', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '216.fh1', 'subgroup_fusion': None, 'weyl_group': '24.14'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 3], 'aut_gens': [[1, 12], [1, 60], [5, 12], [5, 18], [41, 12], [7, 12], [25, 12]], 'aut_group': '96.209', 'aut_hash': 209, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 9, 'aut_perms': [5047, 120, 1, 136, 7, 45360], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 3, 4, 1], [6, 1, 2, 1], [6, 1, 4, 1], [6, 2, 1, 1], [6, 2, 2, 2], [6, 2, 4, 1], [12, 3, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.11', 'autcent_hash': 11, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times D_4', '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, 3], [3, 1, 2], [3, 2, 3], [4, 3, 4], [6, 1, 6], [6, 2, 9], [12, 3, 8]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', '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': 29, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.1', 1], ['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 3, 2, 2], [6, 1, 2, 3], [6, 2, 1, 3], [6, 2, 2, 3], [12, 3, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 12, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '36.12', 'hash': 29, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 60], [25, 12]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 12]], 'label': '72.29', 'linC_count': 96, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 4, 'linQ_dim': 5, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6:C12', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, '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': 14, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 36, 'number_divisions': 20, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 37, 'number_subgroups': 54, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [3, 8], [4, 12], [6, 24], [12, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 18], [41, 12], [11, 12], [7, 12]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [288, 7, 127, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 6]], 'representations': {'PC': {'code': 14483972954469, 'gens': [1, 4], 'pres': [5, -2, -2, -3, -2, -3, 10, 26, 1203, 58, 1204]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [140699802150, 706131050962]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [33562260, 15426728, 34603218, 5426393, 34363418]}, 'Perm': {'d': 12, 'gens': [43908487, 1, 5760, 87091200, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6:C_{12}', 'transitive_degree': 24, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [6, 18, 6, 12, 12], 'aut_gens': [[7176517011515281, 753220439609891, 122003577582205945], [768643328047889283, 1864924272445019, 268987316133934229], [634592314726369681, 4103690531729, 872170512949743239], [743742499327124719, 19625664133349903, 122755322178138865], [622811395350309522, 616450780085944800, 520098798297768766], [622811395393935222, 616450780129560545, 885393716158871545]], 'aut_group': None, 'aut_hash': 5946030899765991394, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 559872, 'aut_permdeg': 216, 'aut_perms': [4213306909311157018148026117595828921987353712341970955642686780984282120658221560806482486072786084954602468580400881312693537179492744727780845183151806877957617725293958941635517582240381781584961883991730896553607774659951738253117678773099462987880648310079811125061247514548910814821289322372305208842587535892356149035741654267374270133922112277274137590901694731980610898186792232966331812769031773896016, 6068956495611174672006533881477803021048333985539013276962094114107514642740345082585724212291559535098180165313338158020841156126183137435899793260477124817397586514897184412368593725657919162473193798202129514016002146393536772978501747930390777022702655499154565570016562078479533046366581237783796679244169971767096722350358337154332584970613886715685226175847620130060017779954610429337976788850866803963113, 1968483204554337085220112190894112765946429858766950729221609519989708400901709263449329769195016300755514719255513571480227247485401342271124785194797034378491123329441446037110715343700284036892989462624321104554413191861111496997321834987289720532033083346097948633681871380917431352177273051298138018096018247253234277610007431078449041065609745762295128697451054956634327627515629232981486936745067818471216, 2751243024575986269070227920100668454567300890827806357931205670359522043732327949798852159655749429186164272778334125118108232152123321511316242002194408969247320745069112735898947843894677312542547491928188561684128506585589163353998531650189886461867784208465660351241796635705572028699152271318725144652301409241066234454076895672191119992569871784231828679713717357225942012358061057845315751629110302397840, 2884644122217545362666741254171634552962155787922445825278367799481995067066940535733542138030471856009680955093159199250615257341841121710219026689332302534038184856597059489337136875439047440543188743465769677829140531628034624585793757955583506980305089293893723768449868370567457508304957327200237456930224166713156918405264128747326015526856071118322972366217201471514562183827303981672194056415174642719648], 'aut_phi_ratio': 108.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 2], [2, 108, 1, 2], [2, 324, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 1], [3, 6, 3, 1], [3, 12, 1, 1], [3, 12, 3, 1], [3, 288, 3, 1], [4, 108, 1, 1], [4, 216, 1, 1], [4, 324, 1, 2], [4, 648, 1, 1], [6, 6, 1, 3], [6, 6, 2, 2], [6, 6, 3, 1], [6, 6, 6, 1], [6, 12, 1, 7], [6, 12, 2, 7], [6, 12, 3, 3], [6, 12, 4, 2], [6, 12, 6, 5], [6, 12, 12, 2], [6, 216, 1, 2], [6, 216, 3, 1], [6, 648, 1, 1], [6, 864, 3, 1], [9, 576, 3, 1], [12, 216, 1, 1], [12, 216, 2, 1], [12, 216, 3, 1], [12, 216, 6, 1], [12, 648, 1, 2], [12, 648, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times C_6^3).C_3^4.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': 36, 'autcentquo_group': None, 'autcentquo_hash': 5946030899765991394, 'autcentquo_nilpotent': False, 'autcentquo_order': 559872, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(C_2\\times C_6^3).C_3^4.C_2^4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 6, 2], [2, 108, 2], [2, 324, 1], [3, 2, 2], [3, 4, 1], [3, 6, 4], [3, 12, 4], [3, 288, 3], [4, 108, 1], [4, 216, 1], [4, 324, 2], [4, 648, 1], [6, 6, 16], [6, 12, 92], [6, 216, 5], [6, 648, 1], [6, 864, 3], [9, 576, 3], [12, 216, 12], [12, 648, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '15552.fa', 'commutator_count': 1, 'commutator_label': None, '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', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 131, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 2], [2, 108, 1, 2], [2, 324, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 4], [3, 12, 1, 4], [3, 288, 1, 3], [4, 108, 1, 1], [4, 216, 1, 1], [4, 324, 1, 2], [4, 648, 1, 1], [6, 6, 1, 6], [6, 6, 2, 5], [6, 12, 1, 26], [6, 12, 2, 33], [6, 216, 1, 5], [6, 648, 1, 1], [6, 864, 1, 3], [9, 576, 1, 3], [12, 216, 1, 4], [12, 216, 2, 4], [12, 648, 1, 2], [12, 648, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2177280, 'exponent': 36, 'exponents_of_order': [6, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 30], [12, 1, 6]], 'familial': False, 'frattini_label': '36.14', 'frattini_quotient': '432.747', 'hash': 8356426295325695289, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [9, 6, 2], 'inner_gens': [[7176517011515281, 19625664133319364, 268987316134014874], [19249053919042422, 753220439609891, 123155638744048646], [372115915581453222, 1864924316000672, 122003577582205945]], 'inner_hash': 8356426295325695289, 'inner_nilpotent': False, 'inner_order': 15552, 'inner_split': True, 'inner_tex': 'C_6^4:D_6', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 10], [3, 4], [4, 4], [6, 44], [12, 96]], 'label': '15552.fa', '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': 'C6^4:D6', 'ngens': 3, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 66, 'number_characteristic_subgroups': 35, 'number_conjugacy_classes': 162, 'number_divisions': 119, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 2197, 'number_subgroup_classes': 4249, 'number_subgroups': 168672, 'old_label': None, 'order': 15552, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 555], [3, 944], [4, 1620], [6, 5520], [9, 1728], [12, 5184]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[7176516931925868, 753220439912809, 122003577582205945], [372115915537961081, 1864924236127200, 122003577582205945]], 'outer_group': '36.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 36, 'outer_permdeg': 8, 'outer_perms': [750, 5761], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times S_3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [3, 4], [4, 4], [6, 24], [12, 40], [24, 33]], 'representations': {'PC': {'code': '24225404114133872460718791277082154162441058376322509748110227502505438953043924831098122685965812199818260453789300045812138', 'gens': [1, 2, 4, 5, 7, 9, 11], 'pres': [11, 2, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 19493, 56, 28778, 1059, 13214, 5701, 9904, 321765, 39307, 158, 9509, 198215, 26604, 226, 44370, 3736, 705680, 513235, 106961, 294, 570249, 285140, 953974, 235245, 117655]}, 'Perm': {'d': 20, 'gens': [7176517011515281, 753220439609891, 122003577582205945]}}, 'schur_multiplier': [2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^4:D_6', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
