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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '832.236', 'ambient_counter': 236, 'ambient_order': 832, 'ambient_tex': 'D_{52}.C_8', 'central': False, 'central_factor': False, 'centralizer_order': 104, 'characteristic': False, 'core_order': 13, 'counter': 30, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '832.236.32.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '32.b1.a1', 'outer_equivalence': False, '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': 32, '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': '26.2', 'subgroup_hash': 2, 'subgroup_order': 26, 'subgroup_tex': 'C_{26}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '832.236', 'aut_centralizer_order': 208, 'aut_label': '32.b1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '12.2', 'aut_weyl_index': 416, 'centralizer': '8.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['16.a1.a1'], 'contains': ['64.a1.a1', '416.b1.a1'], 'core': '64.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [747, -1, 730, -1, 1140, -1, 1140, -1], 'generators': [416, 64], 'label': '832.236.32.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '16.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.a1', 'old_label': '32.b1.a1', 'projective_image': '832.236', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.b1.a1', 'subgroup_fusion': None, 'weyl_group': '4.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '26.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 12, 'aut_gen_orders': [12], 'aut_gens': [[1], [15]], 'aut_group': '12.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 12, 'aut_permdeg': 7, 'aut_perms': [867], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [13, 1, 12, 1], [26, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{12}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 12, 'autcent_group': '12.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{12}', '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], [2, 1, 1], [13, 1, 12], [26, 1, 12]], 'center_label': '26.2', 'center_order': 26, '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': ['2.1', '13.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['13.1', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [13, 1, 12, 1], [26, 1, 12, 1]], 'element_repr_type': 'PC', 'elementary': 26, 'eulerian_function': 1, 'exponent': 26, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 13], 'faithful_reps': [[1, 0, 12]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '26.2', 'hash': 2, 'hyperelementary': 26, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 2, 'irrep_stats': [[1, 26]], 'label': '26.2', 'linC_count': 12, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 1, 'linQ_dim': 12, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C26', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 26, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 26, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [13, 12], [26, 12]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [12], 'outer_gen_pows': [0], 'outer_gens': [[15]], 'outer_group': '12.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [867], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 13], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [12, 2]], 'representations': {'PC': {'code': 731, 'gens': [1], 'pres': [2, -2, -13, 4]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [2211, 26376]}, 'Perm': {'d': 15, 'gens': [87178291200, 5748019200]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [26], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{26}', 'transitive_degree': 26, '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': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 156, 'aut_gen_orders': [12, 4, 6, 12, 4], 'aut_gens': [[1, 2, 32], [17, 714, 544], [441, 370, 688], [17, 618, 304], [1, 586, 96], [441, 690, 816]], 'aut_group': None, 'aut_hash': 6040979354177336406, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4992, 'aut_permdeg': 108, 'aut_perms': [839273594272861477861235196578117156077786657373997946826787020477229087614771189728863757354406786115193770791209174943796703768232238880824594592634581190945404294160423493, 777551161267500297277535575391461054180447274952939227444311656362828185300327756863313841589824709225820852895149478594093882858099648648221131397007372997292069629038609092, 1035441061838439571366572385020417451801312026370681590905275340748393565718873168968716306276600962580297900396777156459537694229106829294800479430892206031997324659085623607, 135054229636944171483820441904938145183320867004362526828369453842049722260785375629895445450775943960714633664349236806031548016950297675818556251719874862421173537076987949, 593573265690833614113823249677495508654733381937123871307339684864283624244082263058272127454695905475871562310145576685078797601858140613322564228816299401560612924287059021], 'aut_phi_ratio': 13.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 52, 1, 1], [4, 1, 1, 2], [4, 2, 1, 1], [4, 52, 1, 1], [8, 4, 1, 2], [8, 13, 2, 2], [8, 26, 1, 2], [13, 4, 3, 1], [16, 26, 4, 2], [16, 52, 2, 2], [26, 4, 3, 1], [26, 8, 3, 1], [52, 4, 3, 2], [52, 8, 3, 1], [104, 8, 6, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_{13}.(C_4^2\\times C_{12}).C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 156, 'autcentquo_group': '624.239', 'autcentquo_hash': 239, 'autcentquo_nilpotent': False, 'autcentquo_order': 624, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_{13}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 52, 1], [4, 1, 2], [4, 2, 1], [4, 52, 1], [8, 4, 2], [8, 13, 4], [8, 26, 2], [13, 4, 3], [16, 26, 8], [16, 52, 4], [26, 4, 3], [26, 8, 3], [52, 4, 6], [52, 8, 3], [104, 8, 12]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '208.34', 'commutator_count': 1, 'commutator_label': '52.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '13.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 236, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 52, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 52, 1, 1], [8, 4, 2, 1], [8, 13, 4, 1], [8, 26, 2, 1], [13, 4, 3, 1], [16, 26, 8, 1], [16, 52, 4, 1], [26, 4, 3, 1], [26, 8, 3, 1], [52, 4, 6, 1], [52, 8, 3, 1], [104, 8, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 208, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 13], 'faithful_reps': [[8, 0, 6]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '104.12', 'hash': 236, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 52, 'inner_gen_orders': [2, 4, 26], 'inner_gens': [[1, 426, 48], [441, 2, 160], [17, 706, 32]], 'inner_hash': 34, 'inner_nilpotent': False, 'inner_order': 208, 'inner_split': True, 'inner_tex': 'D_{26}:C_4', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 192, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 12], [4, 24], [8, 6]], 'label': '832.236', 'linC_count': 192, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 28, 'linQ_degree_count': 2, 'linQ_dim': 28, 'linQ_dim_count': 2, 'linR_count': 48, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D52.C8', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 26, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 58, 'number_divisions': 18, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 56, 'number_subgroups': 442, 'old_label': None, 'order': 832, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 55], [4, 56], [8, 112], [13, 12], [16, 416], [26, 36], [52, 48], [104, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [424, 0], 'outer_gens': [[425, 2, 800], [1, 26, 352]], 'outer_group': '24.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [41064, 1707], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{12}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 45, 'pgroup': 0, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 3], [12, 2], [16, 1], [24, 3], [48, 1]], 'representations': {'PC': {'code': 7004823174071531417395867785526673004593073438731, 'gens': [1, 2, 6], 'pres': [7, -2, -2, -2, -2, -2, 2, -13, 168, 5965, 36, 422, 58, 80, 2021, 3372, 8419, 124, 7853, 9428]}, 'Perm': {'d': 45, 'gens': [2791959419654185476529438605648360077234652483340190747, 5684968252272618161771210935169921358689438132170752000, 8411914642486096081383895494805991489607483551428408702, 11132268982509498343908070775530091815089147228033024000, 13790570011963546347280523430757151532182813503900723200, 16510699386481278262913501252788257969000550203882547200, 613413282]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{52}.C_8', 'transitive_degree': 416, 'wreath_data': None, 'wreath_product': False}
