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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '448.340', 'ambient_counter': 340, 'ambient_order': 448, 'ambient_tex': 'Q_8:D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 56, 'counter': 19, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '448.340.4.k1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '4.k1.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': 4, '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': '112.27', 'subgroup_hash': 27, 'subgroup_order': 112, 'subgroup_tex': 'C_{14}:Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '448.340', 'aut_centralizer_order': 8, 'aut_label': '4.k1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '672.1266', 'aut_weyl_index': 16, 'centralizer': '112.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['8.c1.a1', '8.g1.a1', '8.i1.a1', '8.s1.a1', '8.s1.b1', '8.s1.c1', '8.s1.d1', '28.k1.a1'], 'core': '8.c1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [3499, -1, 8110, -1, 6671, -1, 6067, -1], 'generators': [393, 224, 112, 310, 32], 'label': '448.340.4.k1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.a1', 'old_label': '4.k1.a1', 'projective_image': '112.31', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.k1.a1', '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': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 4, 2, 2, 28], 'aut_gens': [[1, 2, 56], [1, 106, 56], [85, 110, 84], [57, 2, 56], [1, 26, 56], [63, 2, 56]], 'aut_group': '2688.dd', 'aut_hash': 8872731775984099151, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2688, 'aut_permdeg': 15, 'aut_perms': [199308211658, 543272043452, 199348168320, 88745933372, 294154886209], 'aut_phi_ratio': 56.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 2, 1], [4, 14, 4, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 2, 6, 1], [28, 2, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2\\times F_7', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '84.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 84, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 14, 4], [7, 2, 3], [14, 2, 9], [28, 2, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '28.3', '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': 27, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['56.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 14, 1, 4], [7, 2, 3, 1], [14, 2, 3, 3], [28, 2, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 168, 'exponent': 28, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '56.12', 'hash': 27, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 14, 1], 'inner_gens': [[1, 54, 56], [5, 2, 56], [1, 2, 56]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 28, 'inner_split': False, 'inner_tex': 'D_{14}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 26]], 'label': '112.27', 'linC_count': 48, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 10, 'linQ_dim_count': 4, 'linR_count': 48, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C14:Q8', '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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 34, 'number_divisions': 16, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 38, 'number_subgroups': 104, 'old_label': None, 'order': 112, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 60], [7, 6], [14, 18], [28, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [0, 14, 0, 0], 'outer_gens': [[1, 2, 84], [15, 2, 56], [85, 2, 56], [1, 94, 84]], 'outer_group': '96.167', 'outer_hash': 167, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [1144200, 767640, 19441464, 4092484], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 17, '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, 2]], 'representations': {'PC': {'code': 22697803233104895678, 'gens': [1, 2, 5], 'pres': [5, -2, -2, -2, -7, -2, 140, 541, 26, 782, 42, 963]}, 'GLZN': {'d': 2, 'p': 21, 'gens': [9325, 120406, 61776, 80557, 74096]}, 'Perm': {'d': 17, 'gens': [23906015522047, 40320, 45968909051520, 68436399225600, 973]}}, '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': 'C_{14}:Q_8', 'transitive_degree': 112, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [28, 6, 4, 6, 2, 12, 6, 6, 6], 'aut_gens': [[1, 2, 4, 224], [289, 58, 340, 224], [321, 394, 316, 224], [321, 170, 276, 224], [433, 114, 124, 224], [257, 226, 220, 224], [1, 282, 20, 224], [113, 394, 316, 224], [97, 394, 268, 224], [417, 394, 44, 224]], 'aut_group': None, 'aut_hash': 5229004673012084454, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10752, 'aut_permdeg': 44, 'aut_perms': [1752980283193373922121801056817891850822747526726902554, 528328888644097897090802914916637929765898185934059684, 1541394484384176442644145741480210812897822329242836875, 1774990608878792360127178793680540682526870526562191728, 1419539168596364134337954584865052524334870705290232236, 704470441331664466970413306291044454793783972597081264, 987374263962589847277609837294191280668978868690039748, 2616645878642152748488147236178133085437224621790559721, 235079540522079665773954909997097320183289011730490112], 'aut_phi_ratio': 56.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 14, 2, 1], [2, 56, 1, 1], [4, 2, 1, 2], [4, 4, 2, 1], [4, 8, 1, 1], [4, 28, 1, 1], [4, 28, 2, 1], [7, 2, 3, 1], [8, 4, 2, 1], [8, 28, 2, 1], [14, 2, 3, 3], [28, 4, 3, 2], [28, 8, 6, 2], [56, 4, 12, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2^4\\times C_7:C_3).C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 14, 2], [2, 56, 1], [4, 2, 2], [4, 4, 2], [4, 8, 1], [4, 28, 3], [7, 2, 3], [8, 4, 2], [8, 28, 2], [14, 2, 9], [28, 4, 6], [28, 8, 12], [56, 4, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '112.31', 'commutator_count': 1, 'commutator_label': '56.8', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 340, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 14, 1, 2], [2, 56, 1, 1], [4, 2, 1, 2], [4, 4, 1, 2], [4, 8, 1, 1], [4, 28, 1, 3], [7, 2, 3, 1], [8, 4, 2, 1], [8, 28, 2, 1], [14, 2, 3, 3], [28, 4, 3, 2], [28, 8, 3, 2], [28, 8, 6, 1], [56, 4, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2688, 'exponent': 56, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '56.12', 'hash': 340, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 2, 28, 1], 'inner_gens': [[1, 226, 388, 224], [225, 2, 396, 224], [289, 282, 4, 224], [1, 2, 4, 224]], 'inner_hash': 31, 'inner_nilpotent': False, 'inner_order': 112, 'inner_split': None, 'inner_tex': 'D_4\\times D_7', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 34], [4, 19]], 'label': '448.340', 'linC_count': 48, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 48, 'linQ_dim': 12, 'linQ_dim_count': 32, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'Q8:D28', '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': 23, 'number_characteristic_subgroups': 37, 'number_conjugacy_classes': 61, 'number_divisions': 27, 'number_normal_subgroups': 45, 'number_subgroup_autclasses': 124, 'number_subgroup_classes': 158, 'number_subgroups': 980, 'old_label': None, 'order': 448, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 87], [4, 104], [7, 6], [8, 64], [14, 18], [28, 120], [56, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 6], 'outer_gen_pows': [8, 32, 168, 168, 0], 'outer_gens': [[289, 58, 340, 224], [257, 2, 116, 224], [321, 170, 276, 224], [337, 282, 276, 224], [113, 394, 316, 224]], 'outer_group': '96.231', 'outer_hash': 231, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 96, 'outer_perms': [10338163921604963187951458803486319898386901730959792127764094759920324494764567859856557529118014302524616266368225187534432162180425496116551512122, 200212413397059118222421019030207768952336601454744014374745289575252032625561750833682548933971131868007468847884204849763979386103465016673619482670, 31980135039848800582188563608996274542219949684168243083544742995958264608183121698664734218033519171310067305802112096102854148680471511725848711758, 658928742019758671729515121171001696976249754581623996590819753740619671449823835993566725435410746590759500298660527642799024186045045675391935942972, 63303458154253130238040862211332132651746022358719075524294897248819894833104624311285922740264066027290527548746492886046140002925879831303125160847], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 3], [6, 4], [12, 4], [24, 2]], 'representations': {'PC': {'code': 245791066233634231099571229842968905172254, 'gens': [1, 2, 3, 7], 'pres': [7, -2, -2, -2, -2, -2, -7, -2, 3165, 1576, 8150, 4167, 58, 2915, 1690, 80, 7284, 102, 8069]}, 'GLZN': {'d': 2, 'p': 84, 'gens': [12106366, 17588257, 17188445, 17488325, 593713, 49690759, 28499794]}, 'Perm': {'d': 19, 'gens': [357001369764480, 6812, 80199293, 127370880, 13307, 19342, 7116370137676800]}}, 'schur_multiplier': [2, 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:D_{28}', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
