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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '972.463', 'ambient_counter': 463, 'ambient_order': 972, 'ambient_tex': 'C_3^3.S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 9, 'counter': 30, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '972.463.9.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '9.a1.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': 9, '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': '108.27', 'subgroup_hash': 27, 'subgroup_order': 108, 'subgroup_tex': 'C_3:D_{18}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '972.463', 'aut_centralizer_order': 2, 'aut_label': '9.a1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '2916.fs', 'aut_weyl_index': 18, 'centralizer': '486.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1.a1'], 'contains': ['18.b1.a1', '18.j1.a1', '18.s1.a1', '27.a1.a1', '27.b1.a1', '27.b1.b1', '27.b1.c1'], 'core': '108.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [5360, -1, 5951, -1, 4296, -1, 6039, -1], 'generators': [1, 108, 6, 12, 252], 'label': '972.463.9.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '9.a1.a1', 'projective_image': '972.463', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '9.a1.a1', 'subgroup_fusion': None, 'weyl_group': '54.7'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [2, 6, 3, 6, 3, 9], 'aut_gens': [[1, 2, 6], [1, 4, 10], [1, 76, 32], [1, 2, 42], [55, 2, 8], [3, 2, 6], [99, 2, 78]], 'aut_group': '5832.of', 'aut_hash': 6219057450053085502, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5832, 'aut_permdeg': 29, 'aut_perms': [109246259573327459453495305, 1146690238480642926645529996825, 141979080514786944420748393128, 10169659009370258497849, 93724903047811412189677326, 3358365623426763672425789230446], 'aut_phi_ratio': 162.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 27, 2, 1], [3, 2, 1, 1], [3, 2, 3, 1], [6, 2, 1, 1], [6, 2, 3, 1], [9, 2, 9, 1], [18, 2, 9, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_3^4.S_3^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '2916.fs', 'autcentquo_hash': 5014850949365468175, 'autcentquo_nilpotent': False, 'autcentquo_order': 2916, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^4.S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 27, 2], [3, 2, 4], [6, 2, 4], [9, 2, 9], [18, 2, 9]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '54.7', 'commutator_count': 1, 'commutator_label': '27.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 27, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['54.7', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 27, 1, 2], [3, 2, 1, 4], [6, 2, 1, 4], [9, 2, 3, 3], [18, 2, 3, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 84, 'exponent': 18, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '36.13', 'hash': 27, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 3, 9], 'inner_gens': [[1, 4, 102], [5, 2, 6], [13, 2, 6]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 54, 'inner_split': True, 'inner_tex': 'C_3:D_9', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 26]], 'label': '108.27', 'linC_count': 162, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 27, 'linQ_dim': 8, 'linQ_dim_count': 27, 'linR_count': 162, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3:D18', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 30, 'number_divisions': 18, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 50, 'number_subgroups': 248, 'old_label': None, 'order': 108, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 55], [3, 8], [6, 8], [9, 18], [18, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 3, 2, 2, 6], 'outer_gen_pows': [1, 0, 0, 0, 0], 'outer_gens': [[1, 4, 34], [1, 2, 42], [55, 2, 6], [1, 4, 8], [55, 76, 34]], 'outer_group': '108.28', 'outer_hash': 28, 'outer_nilpotent': False, 'outer_order': 108, 'outer_permdeg': 11, 'outer_perms': [21864486, 20704368, 1, 25988665, 1147255], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [6, 6]], 'representations': {'PC': {'code': 8065548553822091, 'gens': [1, 2, 3], 'pres': [5, -2, -3, -2, -3, -3, 41, 1532, 42, 1923, 78, 1804]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [99161, 5869, 61399, 6083, 5941]}, 'GLZq': {'d': 2, 'q': 27, 'gens': [511784, 19765, 203896, 20384, 19927]}, 'Perm': {'d': 14, 'gens': [1118437206, 1, 7784552190, 14545077888, 14545077840]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:D_{18}', 'transitive_degree': 54, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [6, 6, 6], 'aut_gens': [[1, 2, 12, 36, 324], [757, 670, 132, 264, 864], [1, 250, 12, 180, 432], [329, 518, 120, 720, 864]], 'aut_group': None, 'aut_hash': 9143152742808182517, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 52488, 'aut_permdeg': 90, 'aut_perms': [899110331623315633105471872028851227558449859448104605759941483745453982393576822985050158671379929692979783471314161036254686011656853694, 9176266549932044284288954665433711755757464367292942489326944900446581999691364362861331051190372661801900380946687967904300942797057253, 581084583721228075979227468060335947920575896321408714732409741153728935788387374151439916848207485572503014741005378677147757112671265565], 'aut_phi_ratio': 162.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 81, 1, 2], [3, 2, 1, 1], [3, 2, 3, 1], [3, 6, 1, 1], [3, 12, 1, 1], [3, 18, 1, 1], [3, 36, 1, 1], [6, 18, 1, 1], [6, 18, 3, 1], [6, 162, 1, 2], [9, 6, 9, 1], [9, 36, 3, 1], [18, 18, 9, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^4.C_3^4.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': None, 'autcentquo_hash': 9143152742808182517, 'autcentquo_nilpotent': False, 'autcentquo_order': 52488, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^4.C_3^4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 81, 2], [3, 2, 4], [3, 6, 1], [3, 12, 1], [3, 18, 1], [3, 36, 1], [6, 18, 4], [6, 162, 2], [9, 6, 9], [9, 36, 3], [18, 18, 9]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '972.463', 'commutator_count': 1, 'commutator_label': '243.52', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 463, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 81, 1, 2], [3, 2, 1, 4], [3, 6, 1, 1], [3, 12, 1, 1], [3, 18, 1, 1], [3, 36, 1, 1], [6, 18, 1, 4], [6, 162, 1, 2], [9, 6, 3, 3], [9, 36, 1, 3], [18, 18, 3, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6048, 'exponent': 18, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '108.39', 'hash': 463, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 6, 3, 9, 3], 'inner_gens': [[1, 10, 12, 36, 864], [5, 2, 24, 828, 864], [1, 26, 12, 36, 324], [1, 506, 12, 36, 324], [757, 758, 12, 36, 324]], 'inner_hash': 463, 'inner_nilpotent': False, 'inner_order': 972, 'inner_split': True, 'inner_tex': 'C_3^3.S_3^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 10], [4, 4], [6, 20], [12, 1]], 'label': '972.463', 'linC_count': 108, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 36, 'linQ_dim': 20, 'linQ_dim_count': 36, 'linR_count': 108, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3.S3^2', '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': 17, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 39, 'number_divisions': 27, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 125, 'number_subgroup_classes': 197, 'number_subgroups': 3388, 'old_label': None, 'order': 972, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 171], [3, 80], [6, 396], [9, 162], [18, 162]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 3, 3, 2], 'outer_gen_pows': [6, 0, 0, 0], 'outer_gens': [[1, 2, 24, 84, 648], [5, 2, 228, 360, 432], [9, 2, 12, 792, 540], [1, 2, 24, 36, 324]], 'outer_group': '54.8', 'outer_hash': 8, 'outer_nilpotent': False, 'outer_order': 54, 'outer_permdeg': 9, 'outer_perms': [234613, 53286, 123858, 2356], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:S_3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 4], [6, 2], [12, 1], [18, 6]], 'representations': {'PC': {'code': 77621708580478043507583229523042439031071, 'gens': [1, 2, 4, 5, 7], 'pres': [7, -2, -2, -3, -3, -3, -3, 3, 141, 36, 170, 346, 14501, 6003, 137, 4548, 42342, 21181, 6635]}, 'Perm': {'d': 30, 'gens': [3242503207276907033357337120, 994518664721293858296646693921, 10106889855832851195545968251603, 19287152569427005586593577664868, 1843203592011261850147036144320, 28750406600475060344402296122240, 19287152569427005586593577664864]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3.S_3^2', 'transitive_degree': 81, 'wreath_data': None, 'wreath_product': False}
