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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1568.847', 'ambient_counter': 847, 'ambient_order': 1568, 'ambient_tex': 'C_2.D_{14}^2', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 56, 'counter': 28, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1568.847.14.g1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '14.g1', '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': 14, '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': '112.33', 'subgroup_hash': 33, 'subgroup_order': 112, 'subgroup_tex': 'Q_8\\times D_7', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1568.847', 'aut_centralizer_order': 12, 'aut_label': '14.g1', 'aut_quo_index': None, 'aut_stab_index': 112, 'aut_weyl_group': '672.1093', 'aut_weyl_index': 1344, 'centralizer': '392.a1', 'complements': None, 'conjugacy_class_count': 16, 'contained_in': ['2.e1', '7.a1'], 'contains': ['28.c1', '28.j1', '28.l1', '28.m1', '28.n1', '98.e1'], 'core': '28.c1', 'coset_action_label': None, 'count': 112, 'diagramx': [7812, -1, 7889, -1], 'generators': [1, 392, 784, 2, 224], 'label': '1568.847.14.g1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.e1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1', 'old_label': '14.g1', 'projective_image': '784.169', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '14.g1', 'subgroup_fusion': None, 'weyl_group': '56.12'}
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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': 3, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 6], 'aut_gens': [[1, 2, 4], [1, 2, 76], [28, 50, 73], [28, 90, 101]], 'aut_group': '2016.dc', 'aut_hash': 2868159204135431305, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2016, 'aut_permdeg': 13, 'aut_perms': [3669724, 1037878571, 1528493529], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 2, 1], [4, 2, 3, 1], [4, 14, 3, 1], [7, 2, 3, 1], [14, 2, 3, 1], [28, 4, 9, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_4\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 7, 2], [4, 2, 3], [4, 14, 3], [7, 2, 3], [14, 2, 3], [28, 4, 9]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '56.12', 'commutator_count': 1, 'commutator_label': '14.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 33, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 1, 2], [4, 2, 1, 3], [4, 14, 1, 3], [7, 2, 3, 1], [14, 2, 3, 1], [28, 4, 3, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 224, 'exponent': 28, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, -1, 3]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '56.12', 'hash': 33, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 2, 14], 'inner_gens': [[1, 2, 60], [1, 2, 52], [57, 66, 4]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': True, 'inner_tex': 'C_2\\times D_{14}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 24, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 14], [4, 3]], 'label': '112.33', 'linC_count': 27, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 8, 'linQ_dim': 10, 'linQ_dim_count': 8, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'Q8*D7', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 25, 'number_divisions': 15, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 38, 'number_subgroups': 116, 'old_label': None, 'order': 112, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 15], [4, 48], [7, 6], [14, 6], [28, 36]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [1, 0], 'outer_gens': [[1, 2, 69], [28, 58, 61]], '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': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [6, 4], [12, 1]], 'representations': {'PC': {'code': 1270878112850725435109, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -7, 280, 902, 397, 42, 1048, 58, 1209]}, 'GLZN': {'d': 2, 'p': 21, 'gens': [9325, 126881, 61776, 80557, 74096]}, 'Perm': {'d': 15, 'gens': [6267305809, 12460, 18523, 5329, 99712529280]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_8\\times D_7', 'transitive_degree': 56, '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': 3, 'aut_exponent': 84, 'aut_gen_orders': [12, 6, 6, 6, 28, 12, 12], 'aut_gens': [[1, 2, 4, 56], [807, 386, 1484, 424], [1205, 834, 52, 168], [589, 586, 52, 1400], [1233, 494, 44, 728], [39, 374, 364, 1228], [835, 694, 1260, 1196], [1, 14, 828, 308]], 'aut_group': None, 'aut_hash': 8829740922017707739, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 903168, 'aut_permdeg': 112, 'aut_perms': [53319834826267575512171683867758868567312177798024935947701688165187210680869943305050666574809976865673778619352800249270041473762808758388842321575603861744005347760319705362015628, 81820530351837225133290923558454029665356824766860754441876626666058243505896543778706385551584814496437101164561566623899700815205188518772857213239392917574455788417836615921642419, 151369001143084664019595640694376669910150648642522392517095834898486249235076315395911757960742863024691144874981310683277426466755763441861276319844695177039041883515832050858236612, 183055176336500654960586589481519118353549068156481052929293084482024722686755273809319707855127533343734183917736088948189061526580353438488113569286377063278105435023476956919284673, 190550458614554644734627248075220545884886867140789846393545241123697222154284531403694651172917113377848582162787649205635080624683144297330058247216378088012895362170932542924829520, 143594384039068430705854285567063228675497059472236511493112653135856712833470358102215976410324704836228594728362162497222640048335478708385816814247094545147195473259026618626592089, 1198562970863591131413739995269368677688308422027070738801554707211076780726416269636617850428960879846022035781393838807230707903313602800545553293501451085507852165060314143575340], 'aut_phi_ratio': 1344.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 49, 4, 1], [4, 2, 2, 1], [4, 14, 8, 1], [4, 98, 2, 1], [7, 2, 6, 1], [7, 4, 9, 1], [14, 2, 6, 1], [14, 2, 12, 1], [14, 4, 9, 1], [14, 4, 18, 1], [28, 4, 12, 1], [28, 4, 36, 1], [28, 28, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_7:D_7:C_3.C_2^4.C_6.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '7056.bg', 'autcentquo_hash': 7944870934666851665, 'autcentquo_nilpotent': False, 'autcentquo_order': 7056, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_7^2:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 49, 4], [4, 2, 2], [4, 14, 8], [4, 98, 2], [7, 2, 6], [7, 4, 9], [14, 2, 18], [14, 4, 27], [28, 4, 48], [28, 28, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '392.41', 'commutator_count': 1, 'commutator_label': '98.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '7.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 847, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['784.121', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 49, 1, 4], [4, 2, 1, 2], [4, 14, 1, 8], [4, 98, 1, 2], [7, 2, 3, 2], [7, 4, 3, 3], [14, 2, 3, 6], [14, 4, 3, 9], [28, 4, 3, 4], [28, 4, 6, 6], [28, 28, 3, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2046870, 'exponent': 28, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '784.169', 'hash': 847, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 2, 7, 14], 'inner_gens': [[1, 2, 4, 1512], [1, 2, 52, 728], [1, 10, 4, 56], [113, 898, 4, 56]], 'inner_hash': 41, 'inner_nilpotent': False, 'inner_order': 392, 'inner_split': False, 'inner_tex': 'D_7\\times D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 52], [4, 84]], 'label': '1568.847', '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': 'C2.D14^2', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 152, 'number_divisions': 58, 'number_normal_subgroups': 124, 'number_subgroup_autclasses': 90, 'number_subgroup_classes': 393, 'number_subgroups': 4302, 'old_label': None, 'order': 1568, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 199], [4, 312], [7, 48], [14, 144], [28, 864]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [6, 12, 6, 12, 12, 4], 'outer_gen_pows': [0, 0, 2, 0, 0, 0], 'outer_gens': [[29, 814, 12, 532], [3, 2, 1036, 1228], [31, 2, 1372, 1192], [815, 814, 476, 1192], [393, 814, 44, 756], [815, 814, 140, 436]], 'outer_group': None, 'outer_hash': 6889544333894396836, 'outer_nilpotent': False, 'outer_order': 2304, 'outer_permdeg': 22, 'outer_perms': [372746951682308501783, 161186202261478518606, 371694682971401760, 260347191438234701663, 321534364415938249816, 372612857515692917777], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': '(C_6^2\\times D_4).C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4], [6, 16], [12, 16], [24, 6]], 'representations': {'PC': {'code': 26483637328640604887636753813195914625454651697085, 'gens': [1, 2, 3, 5], 'pres': [7, -2, -2, -2, -7, -2, -2, -7, 5488, 555, 58, 682, 52924, 12751, 102, 61157, 30588, 124, 65862, 32941]}, 'Perm': {'d': 24, 'gens': [26064415595284790635566, 25968823152520864958767, 1, 859975, 1275486, 55344551597557255756800, 82376623366764094368000]}}, 'schur_multiplier': [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': 'C_2.D_{14}^2', 'transitive_degree': 112, 'wreath_data': None, 'wreath_product': False}
