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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '512.7530076', 'ambient_counter': 7530076, 'ambient_order': 512, 'ambient_tex': 'D_4^2:C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 8, 'counter': 370, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '512.7530076.16.dr1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '16.dr1', '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': 16, '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': '32.27', 'subgroup_hash': 27, 'subgroup_order': 32, 'subgroup_tex': 'C_2^2\\wr C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '512.7530076', 'aut_centralizer_order': None, 'aut_label': '16.dr1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '64.u1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['8.cd1', '8.cj1', '8.cq1', '8.ct1', '8.cw1', '8.df1', '8.dt1'], 'contains': ['32.cb1', '32.cc1', '32.cg1', '32.ej1', '32.em1', '32.fi1', '32.fn1'], 'core': '64.a1', 'coset_action_label': None, 'count': 8, 'diagramx': [3831, -1, 3830, -1], 'generators': [2789792421136, 6913949046478, 5167, 9703701878542, 1313941673647], 'label': '512.7530076.16.dr1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.g1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.f1', 'old_label': '16.dr1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.dr1', 'subgroup_fusion': None, 'weyl_group': None}
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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': 12, 'aut_gen_orders': [2, 2, 2, 3, 2, 2, 2, 2], 'aut_gens': [[233, 235, 195, 231], [67, 235, 73, 231], [99, 227, 195, 239], [107, 97, 195, 101], [235, 67, 203, 207], [225, 97, 195, 231], [99, 105, 195, 101], [233, 235, 195, 109], [233, 235, 195, 101]], 'aut_group': '384.20100', 'aut_hash': 20100, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 12, 'aut_perms': [12785662, 302392930, 147357745, 52098429, 25832449, 125220487, 98166967, 136110610], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 2, 6, 1], [2, 4, 1, 1], [4, 4, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\wr S_3', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 6], [2, 4, 1], [4, 4, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 27, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 6], [2, 4, 1, 1], [4, 4, 1, 3]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 28, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '8.5', 'hash': 27, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 2], 'inner_gens': [[233, 235, 195, 109], [233, 235, 195, 101], [233, 235, 195, 231], [107, 97, 195, 231]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 6]], 'label': '32.27', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 12, 'linQ_dim': 4, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2wrC2', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 14, 'number_divisions': 14, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 19, 'number_subgroup_classes': 65, 'number_subgroups': 106, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 19], [4, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [65, 65, 65, 65, 65], 'outer_gens': [[67, 235, 73, 231], [99, 227, 195, 231], [235, 67, 203, 207], [99, 105, 195, 231], [233, 105, 195, 231]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [136, 316, 430, 269, 82], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6]], 'representations': {'PC': {'code': 18891336, 'gens': [1, 2, 3, 4], 'pres': [5, 2, 2, 2, 2, 2, 243, 248, 58]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7233260, 16563328]}, 'GLFp': {'d': 4, 'p': 2, 'gens': [33837, 18465, 27183, 18659, 34029]}, 'GLZq': {'d': 2, 'q': 4, 'gens': [67, 235, 69, 195, 73]}, 'Perm': {'d': 8, 'gens': [23616, 16687, 11543, 11536, 5167]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\wr C_2', 'transitive_degree': 8, 'wreath_data': ['C_2^2', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [4, 2, 2, 4, 8, 8, 2], 'aut_gens': [[11212821043200, 6920176400047, 1313941685176, 5612462133817, 2789792421136, 5606234750016, 2789832700823, 5167, 1313941673647], [14008840853062, 1313941685176, 5606234731567, 6913949034937, 4103734089616, 4103693809943, 6920176423663, 1313941668480, 1313941673647], [12520495779121, 4103693798400, 23616, 2789792426303, 5612462138742, 4103693809943, 6920176418496, 5167, 1313941673647], [14002613464336, 6920176394880, 1313941680023, 6913949034937, 2789792421136, 6920176423663, 4103693815096, 5167, 1313941673647], [15310288195342, 6920176394880, 16696, 6913949039862, 4103734089616, 6920176418496, 2789832705976, 5167, 1313941673647], [11219048427001, 5606234726400, 16696, 2789792426303, 6913949039862, 5606234750016, 2789832705976, 5167, 1313941673647], [19608848214352, 4103693803567, 1313941692096, 6913949034937, 2789792421136, 2789832705976, 6920176423663, 5167, 1313941673647], [18132997466863, 5606234726400, 1313941685176, 5612462133817, 2789792421136, 5606234750016, 4103693809943, 5167, 1313941673647]], 'aut_group': '16384.ku', 'aut_hash': 6461665870828750133, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 16384, 'aut_permdeg': 48, 'aut_perms': [2714847770522233919875424233329916378546681010225094452987058, 2036583356512394302185676913081984815675660833634498549455431, 8146712296055260091802838208642175025131798634818507680641631, 6270998563231891915627835749423032959071081572525137232621766, 374469862533098215381987692840467528090573576502118035359158, 7021758928045821651260909567162827334414278512250178829607463, 5996098752287112503955625357947504781285347271188855688828744], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 2, 4, 1], [2, 4, 1, 1], [2, 4, 2, 1], [2, 4, 4, 2], [2, 8, 1, 1], [2, 8, 2, 1], [2, 8, 4, 2], [4, 2, 2, 1], [4, 4, 1, 1], [4, 4, 2, 3], [4, 8, 1, 2], [4, 8, 2, 4], [4, 8, 4, 2], [4, 16, 8, 1], [8, 16, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^5.D_4^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '512.6480905', 'autcentquo_hash': 411620650426100056, 'autcentquo_nilpotent': True, 'autcentquo_order': 512, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3:D_4^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 5], [2, 4, 11], [2, 8, 11], [4, 2, 2], [4, 4, 7], [4, 8, 18], [4, 16, 8], [8, 16, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '256.53380', 'commutator_count': 1, 'commutator_label': '16.11', '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', '2.1', '2.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 7530076, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 5], [2, 4, 1, 11], [2, 8, 1, 11], [4, 2, 1, 2], [4, 4, 1, 7], [4, 8, 1, 18], [4, 16, 1, 8], [8, 16, 1, 4]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 639959040, 'exponent': 8, 'exponents_of_order': [9], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[8, 1, 4]], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '32.51', 'hash': 7530076, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 1], 'inner_gens': [[11212821043200, 1313941697263, 4103693803567, 5612462133817, 2789792421136, 5606234750016, 2789832700823, 1313941668480, 1313941673647], [16819055793216, 6920176400047, 1313941685176, 5612462133817, 2789792421136, 5606234750016, 4103693809943, 5167, 1313941673647], [14002653744023, 6920176400047, 1313941685176, 5612462133817, 2789792421136, 5606234755183, 2789832700823, 5167, 1313941673647], [11212821043200, 6920176400047, 1313941685176, 5612462133817, 4103734094783, 5606234750016, 2789832700823, 5167, 1313941673647], [11212821043200, 6920176400047, 1313941685176, 6913949039862, 2789792421136, 5606234750016, 2789832700823, 5167, 1313941673647], [11212821043200, 6920176400047, 1313941680023, 5612462133817, 2789792421136, 5606234750016, 4103693815096, 5167, 1313941673647], [11212821043200, 5606234731567, 1313941685176, 5612462133817, 2789792421136, 6920176423663, 2789832700823, 5167, 1313941673647], [12526762716847, 6920176400047, 1313941685176, 5612462133817, 2789792421136, 5606234750016, 2789832700823, 5167, 1313941673647], [11212821043200, 6920176400047, 1313941685176, 5612462133817, 2789792421136, 5606234750016, 2789832700823, 5167, 1313941673647]], 'inner_hash': 53380, 'inner_nilpotent': True, 'inner_order': 256, 'inner_split': True, 'inner_tex': 'C_2^5:D_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 32], [2, 24], [4, 8], [8, 4]], 'label': '512.7530076', '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': True, 'name': 'D4^2:C2^3', 'ngens': 9, 'nilpotency_class': 4, 'nilpotent': True, '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': 27, 'number_characteristic_subgroups': 44, 'number_conjugacy_classes': 68, 'number_divisions': 68, 'number_normal_subgroups': 568, 'number_subgroup_autclasses': 1108, 'number_subgroup_classes': 4194, 'number_subgroups': 14530, 'old_label': None, 'order': 512, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 143], [4, 304], [8, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 8402254554953, 0, 0, 5167, 0], 'outer_gens': [[16825283177017, 4103693803567, 1313941697263, 5612462133817, 4103734089616, 2789832700823, 5606234750016, 5167, 1313941673647], [11212821043200, 6920176400047, 1313941685176, 4103734094783, 5612462133817, 5606234750016, 2789832700823, 5167, 1313941673647], [11212821043200, 6920176400047, 11543, 5612462133817, 2789792421136, 5606234750016, 2789832700823, 5167, 1313941673647], [14002613464336, 6920176400047, 1313941685176, 5612462138742, 2789792421136, 5606234750016, 2789832700823, 5167, 1313941673647], [19615075603078, 6920176400047, 1313941685176, 5612462138742, 2789792426303, 5606234750016, 2789832700823, 5167, 1313941673647], [11212821043200, 5606234726400, 11543, 5612462133817, 2789792421136, 5606234750016, 2789832700823, 5167, 1313941673647]], 'outer_group': '64.226', 'outer_hash': 226, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 16, 'outer_perms': [1328391572785, 87179097602, 4396538203938, 2966658509284, 6723108415325, 14199681472674], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4^2', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 24], [4, 8], [8, 4]], 'representations': {'PC': {'code': '11237655691807102279812964071161329411556008', 'gens': [1, 2, 3, 4, 6, 8], 'pres': [9, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1838, 444, 102, 2408, 365, 158, 17296, 1474, 1771, 214, 12977, 3275]}, 'Perm': {'d': 16, 'gens': [11212821043200, 6920176400047, 1313941685176, 5612462133817, 2789792421136, 5606234750016, 2789832700823, 5167, 1313941673647]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4^2:C_2^3', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
