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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '486.260', 'ambient_counter': 260, 'ambient_order': 486, 'ambient_tex': 'C_3^4:S_3', 'central': False, 'central_factor': False, 'centralizer_order': 243, 'characteristic': False, 'core_order': 9, 'counter': 8, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '486.260.54.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '54.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '54.14', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 14, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 54, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3^2:S_3', 'simple': False, 'solvable': True, 'special_labels': ['C4'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '9.2', 'subgroup_hash': 2, 'subgroup_order': 9, 'subgroup_tex': 'C_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '486.260', 'aut_centralizer_order': 1989715104, 'aut_label': '54.a1', 'aut_quo_index': 1, 'aut_stab_index': 1210, 'aut_weyl_group': '48.29', 'aut_weyl_index': 2407555275840, 'centralizer': '2.a1', 'complements': ['9.a1'], 'conjugacy_class_count': 1210, 'contained_in': ['18.a1', '27.a1'], 'contains': ['162.a1'], 'core': '54.a1', 'coset_action_label': None, 'count': 1210, 'diagramx': [3611, 5000, 3349, 5000], 'generators': [2351581309, 191484025], 'label': '486.260.54.a1', 'mobius_quo': 3, 'mobius_sub': 729, 'normal_closure': '54.a1', 'normal_contained_in': ['18.a1'], 'normal_contains': ['162.a1'], 'normalizer': '1.a1', 'old_label': '54.a1', 'projective_image': '486.260', 'quotient_action_image': '2.1', 'quotient_action_kernel': '27.5', 'quotient_action_kernel_order': 27, 'quotient_fusion': None, 'short_label': '54.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 3], [1, 7], [4, 3]], 'aut_group': '48.29', 'aut_hash': 29, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 8, 'aut_perms': [31834, 28334], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', '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], [3, 1, 8]], 'center_label': '9.2', 'center_order': 9, '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': ['3.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '9.2', 'hash': 2, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 9]], 'label': '9.2', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 6, 'linQ_dim': 4, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 9, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 9, 'order_factorization_type': 2, 'order_stats': [[1, 1], [3, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 7], [4, 3]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [31834, 28334], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 3, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -3, 3]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16858733, 35931237]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [687, 1374]}, 'Perm': {'d': 6, 'gens': [240, 4]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2', 'transitive_degree': 9, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 1132560, 'aut_gen_orders': [121, 2, 3], 'aut_gens': [[1435351114, 2892333419, 2351581309, 25154876, 576898846, 368382181], [2121265556, 2355113313, 1224828790, 25154876, 2218581694, 1383477188], [3310511604, 2892333419, 2351581309, 25154876, 576898846, 2756441483], [1435351114, 2892333419, 2351581309, 2272325038, 576898846, 368382181]], 'aut_group': '115562653240320.a', 'aut_hash': 632204379810423377, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 115562653240320, 'aut_permdeg': 243, 'aut_perms': [20728704531154982206662521097902511754177402768278494075677729284289892425366327596855843177220767607829751256028955690014933080739997550555566714727070152262148167099502170605702110174550404931037061638709515654148761127191201767962708075219195032701006975414419032584848862082250533628235959860238328144223946610438927759372494982426093294467079476033319886380830037975767370306989762608820974761368578456089633874261664687778750611442365830795037966920543751010899719653747, 56586848241311755480324177830747876934255663941799753464590974081145213407850762118057190435535264441369029249766429774619841135240589475958832831045882423500670322784263026519756637746749607888555756867666002071084784537672496008513874582728101222316300366224062834026321980413193847919586444892407783229044351264012883486874074494890259944919317766769461479631049703709179750061411899495019150216457840364876782165190881325062352727415227794158003467845752520668972725774376, 56587165617778856460414566316333672859058005447954158112655509383146035903729620511213614842108640159147312816052480628440187614781941632724127326172788160561720496978431319738904637174728322656982029594530453798326307433128263684744238169932829851420279076028784728148685808298205422527803037001925727493448093962522607142022901980170481070371416323966694399612090578298504115447506792900173096573588570032622671631969408228318715987240442596988451237255386464022984681852401], 'aut_phi_ratio': 713349711360.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 121, 1]], 'aut_supersolvable': False, 'aut_tex': '\\AGL(5,3)', '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': 1132560, 'autcentquo_group': '115562653240320.a', 'autcentquo_hash': 632204379810423377, 'autcentquo_nilpotent': False, 'autcentquo_order': 115562653240320, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\AGL(5,3)', 'cc_stats': [[1, 1, 1], [2, 243, 1], [3, 2, 121]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '486.260', 'commutator_count': 1, 'commutator_label': '243.67', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 260, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 1, 121]], 'element_repr_type': 'GLFq', 'elementary': 1, 'eulerian_function': 63, 'exponent': 6, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 5, 11, 13], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '486.260', 'hash': 260, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 3, 2, 3, 3], 'inner_gens': [[1435351114, 2892333419, 2351581309, 1445625641, 576898846, 368382181], [1435351114, 2892333419, 2351581309, 2888790477, 576898846, 368382181], [1435351114, 2892333419, 2351581309, 2345912606, 576898846, 368382181], [2468342098, 2321716242, 1185357808, 25154876, 1137795295, 191484025], [1435351114, 2892333419, 2351581309, 586110491, 576898846, 368382181], [1435351114, 2892333419, 2351581309, 364839242, 576898846, 368382181]], 'inner_hash': 260, 'inner_nilpotent': False, 'inner_order': 486, 'inner_split': True, 'inner_tex': 'C_3^4:S_3', 'inner_used': [1, 2, 3, 4, 5, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 121]], 'label': '486.260', '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': 'C3^4:S3', 'ngens': 6, '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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 123, 'number_divisions': 123, 'number_normal_subgroups': 2665, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 5328, 'number_subgroups': 56632, 'old_label': None, 'order': 486, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 243], [3, 242]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 1132560, 'outer_gen_orders': [3, 5], 'outer_gen_pows': [25154876, 14348908], 'outer_gens': [[3271394916, 1525020304, 1185357808, 25154876, 350488148, 980207619], [3069386852, 2848079468, 211510383, 25154876, 980207619, 104628271]], 'outer_group': '237783237120.a', 'outer_hash': 3786363380318986680, 'outer_nilpotent': False, 'outer_order': 237783237120, 'outer_permdeg': 121, 'outer_perms': [642918142256503458059786895071656860016602763450765560460182761926889417092061927780677813852000684007776826010790345747415962834908298009276145043588052997583163728419278935428749310397948537655519395, 778851398001434174457157092584816524824863900696212592598860293238764876631262858196023194421496120800146484570038422576952145838237524793804747553128815814661010408453797895640713692477494802296460116], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\SL(5,3)', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 6, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 121]], 'representations': {'PC': {'code': 3767160318631933983, 'gens': [1, 2, 3, 4, 5, 6], 'pres': [6, -2, -3, -3, -3, -3, -3, 49, 218, 867, 3244, 11669]}, 'GLFq': {'d': 2, 'q': 243, 'gens': [1435351114, 2892333419, 2351581309, 25154876, 576898846, 368382181]}, 'Perm': {'d': 15, 'gens': [6230654665, 93448903104, 174400128000, 147, 148, 43545744]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:S_3', 'transitive_degree': 243, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 936, 'aut_gen_orders': [2, 13, 3], 'aut_gens': [[5065, 45360, 45507, 3], [5309, 45604, 45507, 3], [85802, 80788, 147, 80640], [5162, 45360, 45507, 3]], 'aut_group': '303264.a', 'aut_hash': 8091482650397092313, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 303264, 'aut_permdeg': 27, 'aut_perms': [7819566423133406134565757606, 9064104205874073187461228659, 6315258978654224054845415987], 'aut_phi_ratio': 16848.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [3, 2, 13, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3:\\GL(3,3)', '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': 936, 'autcentquo_group': '303264.a', 'autcentquo_hash': 8091482650397092313, 'autcentquo_nilpotent': False, 'autcentquo_order': 303264, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^3:\\GL(3,3)', 'cc_stats': [[1, 1, 1], [2, 27, 1], [3, 2, 13]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '54.14', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [3, 2, 1, 13]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 15, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '54.14', 'hash': 14, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3, 3], 'inner_gens': [[5065, 80640, 80884, 4], [85705, 45360, 45507, 3], [85949, 45360, 45507, 3], [5069, 45360, 45507, 3]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 54, 'inner_split': True, 'inner_tex': 'C_3^2:S_3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 13]], 'label': '54.14', 'linC_count': 234, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 234, 'linQ_dim': 6, 'linQ_dim_count': 234, 'linR_count': 234, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:S3', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 15, 'number_divisions': 15, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 56, 'number_subgroups': 212, 'old_label': None, 'order': 54, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 27], [3, 26]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 312, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [5065, 5065], 'outer_gens': [[5065, 80784, 80788, 148], [5065, 45604, 4, 45508]], 'outer_group': '5616.a', 'outer_hash': 4106865526905646958, 'outer_nilpotent': False, 'outer_order': 5616, 'outer_permdeg': 13, 'outer_perms': [1984925728, 5116349529], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\SL(3,3)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 13]], 'representations': {'PC': {'code': 25289607, 'gens': [1, 2, 3, 4], 'pres': [4, -2, -3, -3, -3, 33, 146, 579]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41715841511539067, 125101736062737953, 41624334336939899, 125101750005088939]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [16278328, 9861463, 11495158, 13069762]}, 'Perm': {'d': 9, 'gens': [5065, 45360, 45507, 3]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:S_3', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
