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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '921984.a', 'ambient_counter': 1, 'ambient_order': 921984, 'ambient_tex': 'D_7\\wr S_4', 'central': False, 'central_factor': False, 'centralizer_order': 98, 'characteristic': False, 'core_order': 1, 'counter': 633, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '921984.a.1344.z1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '1344.z1', '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': 1344, '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': '686.13', 'subgroup_hash': 13, 'subgroup_order': 686, 'subgroup_tex': 'C_7^2:C_{14}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '921984.a', 'aut_centralizer_order': None, 'aut_label': '1344.z1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '9408.bc1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['192.d1', '672.bc1', '672.bf1', '672.bm1', '672.co1', '672.cz1', '672.da1', '672.db1', '672.dd1', '672.df1', '672.di1', '672.dl1', '672.dq1', '672.dr1', '672.ds1', '672.dv1'], 'contains': ['2688.e1', '9408.bq1', '9408.bu1', '9408.bv1', '9408.by1'], 'core': '921984.a1', 'coset_action_label': None, 'count': 12, 'diagramx': None, 'generators': [13499863504, 5122652198518231504, 2080416319680702600140544000, 11308319546007488684359680000], 'label': '921984.a.1344.z1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '48.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '12.d1', 'old_label': '1344.z1', 'projective_image': '921984.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1344.z1', 'subgroup_fusion': None, 'weyl_group': '784.165'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '14.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 336, 'aut_gen_orders': [48, 6, 42], 'aut_gens': [[1, 14, 98], [131, 378, 70], [279, 168, 154], [409, 574, 224]], 'aut_group': None, 'aut_hash': 3504067924914375774, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 592704, 'aut_permdeg': 55, 'aut_perms': [5904842362138246351073828262493878931025643745507689277374248331713283689, 9378241807513355105239512303082340463204809644659270840399501161035905064, 2084848401813622616507850397191176891552211454188027147079469836982247448], 'aut_phi_ratio': 2016.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 49, 1, 1], [7, 1, 6, 1], [7, 2, 24, 1], [7, 2, 144, 1], [14, 49, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6\\times C_7^2:C_3.\\SL(2,7).C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 336, 'autcentquo_group': '98784.a', 'autcentquo_hash': 3516501050887675585, 'autcentquo_nilpotent': False, 'autcentquo_order': 98784, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_7^2.\\GL(2,7)', 'cc_stats': [[1, 1, 1], [2, 49, 1], [7, 1, 6], [7, 2, 168], [14, 49, 6]], 'center_label': '7.1', 'center_order': 7, 'central_product': True, 'central_quotient': '98.4', 'commutator_count': 1, 'commutator_label': '49.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '7.1', '7.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['7.1', 1], ['98.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 49, 1, 1], [7, 1, 6, 1], [7, 2, 3, 8], [7, 2, 6, 24], [14, 49, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 399, 'exponent': 14, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '686.13', 'hash': 13, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7, 7], 'inner_gens': [[1, 84, 588], [29, 14, 98], [197, 14, 98]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 98, 'inner_split': False, 'inner_tex': 'C_7:D_7', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 14], [2, 168]], 'label': '686.13', 'linC_count': 12096, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 224, 'linQ_dim': 18, 'linQ_dim_count': 224, 'linR_count': 3024, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C7^2:C14', '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 182, 'number_divisions': 36, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 88, 'number_subgroups': 328, 'old_label': None, 'order': 686, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 49], [7, 342], [14, 294]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 168, 'outer_gen_orders': [42, 6, 42], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[3, 476, 462], [11, 112, 280], [13, 126, 630]], 'outer_group': None, 'outer_hash': 3846872071170751499, 'outer_nilpotent': False, 'outer_order': 6048, 'outer_permdeg': 16, 'outer_perms': [6993031812508, 550013587204, 6640043564188], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times C_3.\\SO(3,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 7], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [6, 10], [12, 24]], 'representations': {'PC': {'code': 1577289046247, 'gens': [1, 3, 4], 'pres': [4, -2, -7, -7, -7, 8, 1010, 9411]}, 'GLZN': {'d': 2, 'p': 98, 'gens': [53925131, 67101777, 942565, 949815]}, 'GLZq': {'d': 2, 'q': 49, 'gens': [1010178, 117993, 119902, 3496221]}, 'Perm': {'d': 21, 'gens': [2447136481639449727, 5244679414274186880, 7812787262797844173, 973]}}, 'schur_multiplier': [7], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [14], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7^2:C_{14}', 'transitive_degree': 98, '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': 5, 'aut_exponent': 168, 'aut_gen_orders': [12, 12], 'aut_gens': [[22616636880507051861920009519, 304452169334473748238971271936], [92110719186802465071298426275, 90900223223337716562737400239], [192596895989431570039724658153, 191449685806721513377089406043]], 'aut_group': '2765952.c', 'aut_hash': 3126063179686267361, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2765952, 'aut_permdeg': 112, 'aut_perms': [180469817434190323290667342259440030963532313110634032090634412765031309751852032123055454281573103527571007401159095962968805885803204291027452234890345886820836973505854829610482483, 192373813885811045381774136432519808621510156150559916835398478933932785210507619049801885362741237236280029890723506075324231448128331993676986453670898378989357594504222135065300176], 'aut_phi_ratio': 10.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 28, 1, 1], [2, 84, 1, 1], [2, 294, 1, 1], [2, 588, 1, 1], [2, 1176, 1, 1], [2, 1372, 1, 1], [2, 2401, 1, 1], [2, 4116, 1, 1], [3, 1568, 1, 1], [4, 588, 1, 1], [4, 8232, 1, 2], [4, 16464, 1, 1], [4, 28812, 1, 2], [6, 10976, 1, 2], [6, 76832, 1, 1], [7, 8, 3, 1], [7, 16, 3, 1], [7, 24, 3, 1], [7, 32, 3, 1], [7, 48, 3, 1], [7, 64, 3, 2], [7, 96, 3, 3], [7, 192, 1, 1], [7, 192, 3, 1], [8, 115248, 1, 1], [14, 168, 3, 2], [14, 224, 3, 1], [14, 336, 3, 3], [14, 672, 3, 10], [14, 1176, 3, 2], [14, 1344, 1, 1], [14, 1344, 3, 3], [14, 2352, 3, 5], [14, 2744, 3, 1], [14, 4704, 3, 4], [14, 8232, 3, 1], [21, 3136, 3, 2], [21, 6272, 3, 3], [28, 2352, 3, 2], [28, 4704, 3, 1], [28, 16464, 3, 2], [28, 32928, 3, 1], [42, 21952, 3, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_7^4.C_2^3.(C_6\\times S_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': 168, 'autcentquo_group': '2765952.c', 'autcentquo_hash': 3126063179686267361, 'autcentquo_nilpotent': False, 'autcentquo_order': 2765952, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_7^4.C_2^3.(C_6\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 28, 1], [2, 84, 1], [2, 294, 1], [2, 588, 1], [2, 1176, 1], [2, 1372, 1], [2, 2401, 1], [2, 4116, 1], [3, 1568, 1], [4, 588, 1], [4, 8232, 2], [4, 16464, 1], [4, 28812, 2], [6, 10976, 2], [6, 76832, 1], [7, 8, 3], [7, 16, 3], [7, 24, 3], [7, 32, 3], [7, 48, 3], [7, 64, 6], [7, 96, 9], [7, 192, 4], [8, 115248, 1], [14, 168, 6], [14, 224, 3], [14, 336, 9], [14, 672, 30], [14, 1176, 6], [14, 1344, 10], [14, 2352, 15], [14, 2744, 3], [14, 4704, 12], [14, 8232, 3], [21, 3136, 6], [21, 6272, 9], [28, 2352, 6], [28, 4704, 3], [28, 16464, 6], [28, 32928, 3], [42, 21952, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '921984.a', 'commutator_count': 1, 'commutator_label': '230496.h', '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', '7.1', '7.1', '7.1', '7.1'], 'composition_length': 12, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 5, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 28, 1, 1], [2, 84, 1, 1], [2, 294, 1, 1], [2, 588, 1, 1], [2, 1176, 1, 1], [2, 1372, 1, 1], [2, 2401, 1, 1], [2, 4116, 1, 1], [3, 1568, 1, 1], [4, 588, 1, 1], [4, 8232, 1, 2], [4, 16464, 1, 1], [4, 28812, 1, 2], [6, 10976, 1, 2], [6, 76832, 1, 1], [7, 8, 3, 1], [7, 16, 3, 1], [7, 24, 3, 1], [7, 32, 3, 1], [7, 48, 3, 1], [7, 64, 3, 2], [7, 96, 3, 3], [7, 192, 1, 1], [7, 192, 3, 1], [8, 115248, 1, 1], [14, 168, 3, 2], [14, 224, 3, 1], [14, 336, 3, 3], [14, 672, 3, 10], [14, 1176, 3, 2], [14, 1344, 1, 1], [14, 1344, 3, 3], [14, 2352, 3, 5], [14, 2744, 3, 1], [14, 4704, 3, 4], [14, 8232, 3, 1], [21, 3136, 3, 2], [21, 6272, 3, 3], [28, 2352, 3, 2], [28, 4704, 3, 1], [28, 16464, 3, 2], [28, 32928, 3, 1], [42, 21952, 3, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 57600, 'exponent': 168, 'exponents_of_order': [7, 4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, 1, 12], [16, 1, 12], [24, 1, 36], [32, 1, 15], [48, 1, 24], [64, 1, 18], [96, 1, 39], [128, 1, 6], [192, 1, 8]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '921984.a', 'hash': 373698978928536462, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 168, 'inner_gen_orders': [42, 8], 'inner_gens': [[22616636880507051861920009519, 192596896017876917537847109370], [203905213285485569641039874610, 304452169334473748238971271936]], 'inner_hash': 373698978928536462, 'inner_nilpotent': False, 'inner_order': 921984, 'inner_split': True, 'inner_tex': 'D_7\\wr S_4', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 2], [3, 4], [4, 4], [6, 4], [8, 14], [16, 12], [24, 36], [32, 15], [48, 24], [64, 18], [96, 39], [128, 6], [192, 8]], 'label': '921984.a', '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': None, 'name': 'D7wrS4', 'ngens': 2, '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, 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': 78, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 190, 'number_divisions': 78, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 1913, 'number_subgroup_classes': 1913, 'number_subgroups': 5155532, 'old_label': None, 'order': 921984, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 10059], [3, 1568], [4, 91140], [6, 98784], [7, 2400], [8, 115248], [14, 170016], [21, 75264], [28, 225792], [42, 131712]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [2080416331716787251583160399], 'outer_gens': [[33924877755938355967405190063, 259299605647795704453481788000]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': 9, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [3, 4], [4, 4], [6, 4], [8, 2], [24, 4], [48, 4], [72, 12], [96, 5], [144, 8], [192, 8], [288, 13], [384, 2], [576, 2]], 'representations': {'PC': {'code': '16044673525230210845328135585449957508798179033581604631229821391899728936745330113634708679058488454633901760600596243082343675209770046515212377362264159041146460888370997548468856368986004182359242367399353232671028041429488365679727504953869193873966055840637730686200382782523680392133805761520989633390284521585869612884546670952755705488668318861513045836887612051495', 'gens': [1, 2, 4, 5, 6, 8, 9, 11, 12], 'pres': [12, 2, 2, 3, 2, 2, 2, 2, 7, 2, 7, 7, 7, 3350304, 18791377, 61, 31609730, 10394, 22123587, 4005231, 6172731, 79911, 7743604, 4919416, 259768, 4600480, 63603365, 10237553, 15463685, 3550217, 2299013, 209, 66471558, 34211538, 8134590, 492618, 265830, 16551943, 448195, 1231, 87901208, 33067460, 11839316, 673964, 2074952, 1469300, 83240, 11432, 320, 88081929, 41322261, 20574753, 543405, 4233657, 62949, 20241, 13229578, 11392150, 7400482, 623098, 1034950, 155314, 11194, 5861387, 17625623, 3331619, 9709103, 555323, 733895, 1185491, 84779]}, 'Perm': {'d': 28, 'gens': [304452169334473748238971271936, 22616636880507051861920009519]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_7\\wr S_4', 'transitive_degree': 28, 'wreath_data': ['65856.h1', '38416.e1', '153664.b1', '4T5'], 'wreath_product': True}
