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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1728.47319', 'ambient_counter': 47319, 'ambient_order': 1728, 'ambient_tex': 'C_{12}:D_6^2', 'central': False, 'central_factor': False, 'centralizer_order': 576, 'characteristic': False, 'core_order': 4, 'counter': 696, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.47319.216.g1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '216.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': 216, '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': '8.5', 'subgroup_hash': 5, 'subgroup_order': 8, 'subgroup_tex': 'C_2^3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.47319', 'aut_centralizer_order': None, 'aut_label': '216.g1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '3.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['72.a1', '72.v1', '72.w1', '72.x1', '108.b1', '108.c1', '108.d1', '108.e1', '108.f1'], 'contains': ['432.a1', '432.c1', '432.h1'], 'core': '432.a1', 'coset_action_label': None, 'count': 6, 'diagramx': None, 'generators': [289675, 6612431, 12362371], 'label': '1728.47319.216.g1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '72.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '216.g1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '216.g1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 84, 'aut_gen_orders': [3, 3], 'aut_gens': [[1, 2, 4], [4, 5, 3], [2, 4, 1]], 'aut_group': '168.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 7, 'aut_perms': [4361, 244], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 168, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\PSL(2,7)', '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, 7]], 'center_label': '8.5', 'center_order': 8, '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', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8]], 'label': '8.5', 'linC_count': 28, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 28, 'linQ_dim': 3, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3', 'ngens': 3, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 8, 'number_divisions': 8, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 5, 3], [2, 4, 1]], 'outer_group': '168.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 168, 'outer_permdeg': 7, 'outer_perms': [4361, 244], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -2, 2, 2]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16482, 16322, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 13286, 13933]}, 'Perm': {'d': 6, 'gens': [120, 6, 1]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3', 'transitive_degree': 8, '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [12, 12, 12, 12, 12, 12, 12, 4], 'aut_gens': [[287519, 289675, 6758390, 291235, 7423033], [431245, 12362393, 18690209, 3125431, 5518687], [18688735, 18689483, 9345863, 3766123, 3793705], [6612409, 3020896, 18833198, 473395, 10118639], [12362393, 9345830, 16245725, 9151441, 7423033], [290423, 16245758, 6758390, 5518549, 11843621], [288971, 289675, 18833198, 12469777, 9148015], [290423, 12364549, 9345863, 10118117, 5518687], [6756179, 287519, 6615335, 434224, 5518687]], 'aut_group': '11664.ic', 'aut_hash': 8868945092102083686, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 442368, 'aut_permdeg': 48, 'aut_perms': [1957390381613245206722214231271819929099541850896478585614281, 4624037928979963158808239390123307562634126223114462870972370, 4112721678553668879017539469411041452042487235732500508925030, 4007940359656182291511820232651588443548661584734788703414695, 5333246862115034450744146599529488928075677785828315044824763, 9587197544092557187315012620256628371481619083465363181515056, 5850009354443124549426955008239009970523579344152318589753915, 68264418377433343956791746846326558267226999966119972486995], 'aut_phi_ratio': 768.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 3, 8, 1], [2, 6, 4, 1], [2, 9, 4, 1], [2, 18, 8, 1], [2, 54, 4, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [4, 2, 2, 1], [4, 6, 4, 1], [4, 18, 2, 1], [6, 2, 1, 1], [6, 2, 2, 2], [6, 2, 4, 1], [6, 4, 1, 1], [6, 4, 2, 2], [6, 4, 4, 1], [6, 6, 8, 2], [6, 8, 1, 1], [6, 8, 2, 1], [6, 12, 8, 2], [6, 18, 4, 1], [6, 24, 4, 1], [6, 36, 8, 1], [12, 2, 4, 1], [12, 4, 4, 1], [12, 4, 8, 1], [12, 6, 8, 1], [12, 8, 2, 1], [12, 8, 4, 1], [12, 12, 4, 1], [12, 12, 8, 1], [12, 18, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4:(S_3\\times D_{12})', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '512.6249567', 'autcent_hash': 6452065007057432321, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '864.4686', 'autcentquo_hash': 4686, 'autcentquo_nilpotent': False, 'autcentquo_order': 864, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^3:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 8], [2, 6, 4], [2, 9, 4], [2, 18, 8], [2, 54, 4], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 2, 2], [4, 6, 4], [4, 18, 2], [6, 2, 9], [6, 4, 9], [6, 6, 16], [6, 8, 3], [6, 12, 16], [6, 18, 4], [6, 24, 4], [6, 36, 8], [12, 2, 4], [12, 4, 12], [12, 6, 8], [12, 8, 6], [12, 12, 12], [12, 18, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '432.759', 'commutator_count': 1, 'commutator_label': '54.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 47319, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['24.6', 1], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 8], [2, 6, 1, 4], [2, 9, 1, 4], [2, 18, 1, 8], [2, 54, 1, 4], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 2, 1, 2], [4, 6, 1, 4], [4, 18, 1, 2], [6, 2, 1, 9], [6, 4, 1, 9], [6, 6, 1, 16], [6, 8, 1, 3], [6, 12, 1, 16], [6, 18, 1, 4], [6, 24, 1, 4], [6, 36, 1, 8], [12, 2, 2, 2], [12, 4, 1, 4], [12, 4, 2, 4], [12, 6, 2, 4], [12, 8, 1, 2], [12, 8, 2, 2], [12, 12, 1, 4], [12, 12, 2, 4], [12, 18, 2, 2]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 9999360000, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4704', 'hash': 47319, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 3, 6, 6], 'inner_gens': [[287519, 289675, 6758390, 289783, 7423033], [287519, 289675, 16245725, 291235, 7423033], [287519, 9920824, 6758390, 291235, 7423033], [290423, 289675, 6758390, 291235, 3793705], [287519, 289675, 6758390, 7216855, 7423033]], 'inner_hash': 759, 'inner_nilpotent': False, 'inner_order': 432, 'inner_split': False, 'inner_tex': 'C_2\\times S_3^3', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 72], [4, 48], [8, 10]], 'label': '1728.47319', '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': 'C12:D6^2', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 42, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 162, 'number_divisions': 144, 'number_normal_subgroups': 780, 'number_subgroup_autclasses': 780, 'number_subgroup_classes': 6210, 'number_subgroups': 57780, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 447], [3, 26], [4, 64], [6, 822], [12, 368]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 2, 4, 4, 4, 2], 'outer_gen_pows': [287497, 287497, 287497, 287497, 287497, 287497], 'outer_gens': [[289675, 12362393, 6613883, 9921625, 13747967], [12362393, 289675, 6758390, 289783, 13747967], [6612409, 289675, 9345863, 289783, 13747967], [18689483, 287519, 6615335, 434224, 7423033], [6612409, 289675, 9345863, 473395, 7423033], [18689483, 18687283, 6615335, 9921625, 7423033]], 'outer_group': None, 'outer_hash': 2734666131921231189, 'outer_nilpotent': True, 'outer_order': 1024, 'outer_permdeg': 16, 'outer_perms': [1307757865363, 6234646636, 2795972638126, 1313909027830, 1314021178741, 1313988514990], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5.C_2^5', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 56], [4, 40], [8, 14], [16, 2]], 'representations': {'PC': {'code': 2140574014706471589534882536529368598341594753065, 'gens': [1, 2, 3, 5, 7], 'pres': [9, -2, -2, -2, -3, -2, -3, -2, -2, -3, 281, 74, 300, 5404, 130, 5189, 4200, 186, 4363, 214, 3932]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [2419376, 6612431, 433456, 12470437, 289675, 287519, 3911542, 288949, 12362371]}, 'Perm': {'d': 15, 'gens': [40284856, 368063, 6267744000, 40284720, 1169280, 367920, 325, 435, 93405312000]}}, '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': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}:D_6^2', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
