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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '384.16394', 'ambient_counter': 16394, 'ambient_order': 384, 'ambient_tex': 'D_{24}:C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 32, 'counter': 17, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '384.16394.6.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '6.b1', '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': 6, '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': '64.186', 'subgroup_hash': 186, 'subgroup_order': 64, 'subgroup_tex': 'C_2\\times D_{16}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.16394', 'aut_centralizer_order': None, 'aut_label': '6.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '48.c1', 'complements': None, 'conjugacy_class_count': 12, 'contained_in': ['2.a1', '3.a1'], 'contains': ['12.a1', '12.d1', '12.f1', '12.i1'], 'core': '12.a1', 'coset_action_label': None, 'count': 36, 'diagramx': None, 'generators': [3, 4, 192], 'label': '384.16394.6.b1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '6.b1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.b1', '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 16, 'aut_gen_orders': [8, 4, 8, 4, 8, 2], 'aut_gens': [[1, 2, 4], [33, 27, 37], [1, 62, 61], [33, 58, 20], [33, 11, 60], [1, 26, 36], [33, 2, 36]], 'aut_group': '1024.dij', 'aut_hash': 598280662459352130, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 1024, 'aut_permdeg': 32, 'aut_perms': [49303369196055213457667724086572906, 177008948780726963880552453850136258, 56990515217765629208286429192000164, 23641155735469361377082517203418395, 48649067522344632591201493561835228, 33169418224095037378996625162619766], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 8, 4, 1], [4, 2, 1, 2], [8, 2, 2, 2], [16, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2^2\\times C_8).C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': '32.43', 'autcentquo_hash': 43, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_8:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 8, 4], [4, 2, 2], [8, 2, 4], [16, 2, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '16.7', 'commutator_count': 1, 'commutator_label': '8.1', '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': 186, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['32.18', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 8, 1, 4], [4, 2, 1, 2], [8, 2, 2, 2], [16, 2, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 84, 'exponent': 16, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '8.5', 'hash': 186, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [1, 2, 8], 'inner_gens': [[1, 2, 4], [1, 2, 60], [1, 10, 4]], 'inner_hash': 7, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': False, 'inner_tex': 'D_8', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 14]], 'label': '64.186', 'linC_count': 32, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 8, 'linQ_dim': 9, 'linQ_dim_count': 8, 'linR_count': 32, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D16', 'ngens': 3, 'nilpotency_class': 4, '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': 9, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 22, 'number_divisions': 14, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 49, 'number_subgroups': 137, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 35], [4, 4], [8, 8], [16, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 4], 'outer_gen_pows': [16, 4, 8, 0], 'outer_gens': [[33, 35, 4], [33, 6, 4], [33, 7, 5], [1, 3, 20]], 'outer_group': '64.206', 'outer_hash': 206, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 12, 'outer_perms': [90842400, 90842423, 261314649, 47991000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2:C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 2], [8, 2]], 'representations': {'PC': {'code': 1180605043711004, 'gens': [1, 2, 3], 'pres': [6, -2, 2, 2, -2, -2, -2, 548, 50, 681, 69, 730, 88]}, 'GLZN': {'d': 2, 'p': 34, 'gens': [687293, 134603, 908763, 1297065, 39883, 39337]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [12339, 4129, 36873, 39119, 4161, 4225]}, 'Perm': {'d': 18, 'gens': [361229975377207, 85740144663967, 816553097098801, 1202407402679670, 1580385119566350, 816553097098800]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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\\times D_{16}', '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': '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': [12, 4, 4, 4, 12, 6], 'aut_gens': [[1, 2, 4, 64], [224, 18, 276, 65], [1, 194, 373, 353], [225, 50, 364, 352], [193, 34, 204, 64], [193, 203, 205, 129], [193, 50, 253, 257]], 'aut_group': None, 'aut_hash': 881124625635868297, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 147456, 'aut_permdeg': 128, 'aut_perms': [280513463258443039541306798340052155963784695123685983991093826398589842513130155678029473861047726557398252085478447607889792762144281871578903279623897081727979760269132521700221293955821046539648754508899708531963, 13089461362153155612509764534039697261029702846094869719011744015154879890596518852514446881672432267302021599201294103524234837630115540443747711082598838333430591377790933470471149245786963423312046979983975754832, 281059646754226179245326062227611137228524392466463888287076806371231060954826074710712382268162615285183151063205279873315745701082458366993939053781787666028951004270023635292555388670692044785946867326807110954551, 19246216549393959981428127041620594911846577286800067466605871047471957325305697140583419689940026285960057245157219276968774257363555463360068794169658916024094380138795254109379400080712734449298249686290767378642, 365907606603718129421704868503133178590466196505833224300067687164289327172729092801738079788705383428957252008590188766971762575677426376165800341329473583451478256012521633215944469992277219183780261227852841538899, 52361941268435377207439588328240014147473569403340879116589072083172630764944265166132245266516188580061712340181710538254256922038104967584055134417431646173910704646723155850467032497836406378450391519694756412744], 'aut_phi_ratio': 1152.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 8, 4, 1], [2, 24, 4, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 2, 3, 1], [6, 2, 1, 1], [6, 2, 6, 1], [6, 8, 8, 1], [8, 2, 2, 1], [8, 2, 6, 1], [12, 4, 1, 1], [12, 4, 3, 1], [16, 6, 16, 1], [24, 4, 2, 1], [24, 4, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6.(C_6\\times S_4).C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': None, 'autcent_hash': 4599629986281706675, 'autcent_nilpotent': False, 'autcent_order': 1536, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4.C_2^4.S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': None, 'autcentquo_hash': 209, 'autcentquo_nilpotent': False, 'autcentquo_order': 96, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times S_3\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 8, 4], [2, 24, 4], [3, 2, 1], [4, 2, 4], [6, 2, 7], [6, 8, 8], [8, 2, 8], [12, 4, 4], [16, 6, 16], [24, 4, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '48.15', 'commutator_count': 1, 'commutator_label': '24.2', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 16394, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['96.33', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 8, 1, 4], [2, 24, 1, 4], [3, 2, 1, 1], [4, 2, 1, 4], [6, 2, 1, 7], [6, 8, 2, 4], [8, 2, 2, 4], [12, 4, 1, 4], [16, 6, 4, 4], [24, 4, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 43680, 'exponent': 48, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '48.51', 'hash': 16394, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [1, 2, 8, 3], 'inner_gens': [[1, 2, 4, 64], [1, 2, 60, 64], [1, 10, 4, 320], [1, 2, 132, 64]], 'inner_hash': 15, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': False, 'inner_tex': 'C_3:D_8', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 44], [4, 12]], 'label': '384.16394', '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': 'D24:C2^3', 'ngens': 8, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 18, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 72, 'number_divisions': 48, 'number_normal_subgroups': 149, 'number_subgroup_autclasses': 108, 'number_subgroup_classes': 440, 'number_subgroups': 1982, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 135], [3, 2], [4, 8], [6, 78], [8, 16], [12, 16], [16, 96], [24, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 4, 12, 4], 'outer_gen_pows': [0, 0, 16, 0], 'outer_gens': [[33, 2, 21, 64], [33, 194, 52, 64], [192, 58, 20, 65], [224, 3, 197, 289]], 'outer_group': None, 'outer_hash': 3982420098773055476, 'outer_nilpotent': False, 'outer_order': 3072, 'outer_permdeg': 22, 'outer_perms': [750861362075638464007, 109621507189562202240, 689378143001986252390, 51731905197854152], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2.C_2^6.C_6.C_2^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 12], [8, 8]], 'representations': {'PC': {'code': 4180357333898798406327819871214593, 'gens': [1, 2, 3, 7], 'pres': [8, -2, -2, -2, -2, -2, -2, -2, -3, 730, 66, 907, 91, 972, 116, 4502, 166, 4119]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [110881, 110623, 4536535, 111745, 5197871, 111361, 1880081, 111169]}, 'Perm': {'d': 23, 'gens': [207686756870005996920, 1134759212496308770567, 16, 2504099257132364680343, 3591734293329888746880, 4820445464152008055680, 2504099257132364680320, 840]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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': 'D_{24}:C_2^3', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
