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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '2304.wi', 'ambient_counter': 581, 'ambient_order': 2304, 'ambient_tex': 'C_3\\times C_2^5:S_4', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 12, 'counter': 104, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2304.wi.24.bc1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '24.bc1', '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': 24, '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': '96.165', 'subgroup_hash': 165, 'subgroup_order': 96, 'subgroup_tex': 'D_4\\times C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2304.wi', 'aut_centralizer_order': 8, 'aut_label': '24.bc1', 'aut_quo_index': None, 'aut_stab_index': 24, 'aut_weyl_group': '64.267', 'aut_weyl_index': 192, 'centralizer': '96.o1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['12.c1', '12.d1', '12.bc1'], 'contains': ['48.bc1', '48.bd1', '48.bg2', '48.bh1', '48.bk1', '48.bl1', '48.bm1', '72.bc1'], 'core': '192.b1', 'coset_action_label': None, 'count': 24, 'diagramx': None, 'generators': [127370880, 174396884245, 93452516635, 40292755, 174356582400, 40318259], 'label': '2304.wi.24.bc1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.g1', 'old_label': '24.bc1', 'projective_image': '768.1090070', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.bc1', 'subgroup_fusion': None, 'weyl_group': '16.14'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '48.44', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 4], 'aut_gens': [[1, 2, 8], [53, 2, 40], [1, 50, 8], [53, 54, 40], [1, 6, 12], [5, 50, 56], [53, 2, 8], [29, 6, 92]], 'aut_group': '256.55643', 'aut_hash': 55643, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 256, 'aut_permdeg': 14, 'aut_perms': [415, 14377673646, 27909186607, 27430179967, 2438997120, 27909186480, 31150708254], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 4, 1], [3, 1, 2, 1], [4, 1, 4, 1], [4, 2, 2, 2], [4, 2, 4, 1], [6, 1, 2, 3], [6, 2, 8, 1], [12, 1, 8, 1], [12, 2, 4, 2], [12, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^5:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [3, 1, 2], [4, 1, 4], [4, 2, 8], [6, 1, 6], [6, 2, 8], [12, 1, 8], [12, 2, 16]], 'center_label': '24.9', 'center_order': 24, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 165, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [3, 1, 2, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 3], [6, 1, 2, 3], [6, 2, 2, 4], [12, 1, 4, 2], [12, 2, 2, 2], [12, 2, 4, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1092, 'exponent': 12, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '24.15', 'hash': 165, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 1, 2], 'inner_gens': [[1, 2, 56], [1, 2, 8], [49, 2, 8]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48], [2, 12]], 'label': '96.165', 'linC_count': 256, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 80, 'linQ_dim': 6, 'linQ_dim_count': 80, 'linR_count': 16, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4*C12', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, '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': 18, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 60, 'number_divisions': 30, 'number_normal_subgroups': 64, 'number_subgroup_autclasses': 52, 'number_subgroup_classes': 94, 'number_subgroups': 124, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 11], [3, 2], [4, 20], [6, 22], [12, 40]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 8, 0], 'outer_gens': [[5, 2, 40], [1, 50, 8], [1, 54, 40], [25, 6, 40], [29, 54, 44]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [1174320, 806520, 1174440, 127, 806538], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 7], [8, 1]], 'representations': {'PC': {'code': 238133119559297, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -2, -2, -2, -3, 31, 1347, 69, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729084381806, 58438769798631180, 91655858515229551]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [297930592, 9155920602, 7226245062, 8360353642, 9789111396, 9098424014]}, 'Perm': {'d': 11, 'gens': [766944, 3670800, 864, 4, 7983360, 1680]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 12], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4\\times C_{12}', 'transitive_degree': 48, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [8, 4, 4, 2, 6, 4], 'aut_gens': [[93408967995, 87123994, 93405331342, 40279680, 127370880, 3758, 174356582400, 174479977677, 174443698781, 93492414105], [93412946062, 127403674, 93405350579, 40292755, 87129779, 18619, 174356582400, 174400145318, 174483978461, 93532685916], [93445268763, 87096690, 93405325075, 127390222, 40292755, 40297565, 174356582400, 174476695913, 174443706394, 93445608129], [93529084804, 40304861, 93405331342, 87091200, 127370880, 3758, 174356582400, 174363879853, 174396867570, 93445627953], [174480355204, 87116381, 174356601742, 40279680, 127370880, 3758, 93405312000, 93485183821, 93492435994, 174443684505], [174396517145, 127396061, 174356595475, 87091200, 40279680, 40297565, 93405312000, 93485149829, 93532688370, 174396880699], [93445234986, 87118015, 93405325075, 127390222, 40292755, 40290585, 174356582400, 174400130518, 174443698781, 93445630989]], 'aut_group': None, 'aut_hash': 1973515413902384372, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 50, 'aut_perms': [29111709452538941485343339225320812330544391295389323410706669740, 23833733728210774345734746439703703093722879296351905375905484454, 26769239940000194849885399882605607610289242190593079086757704246, 616881679258062733554707181661140332868758681649498153843236740, 23829947600039901544280807403036582014784759820725173356652743664, 12104273940326475892161601418153722427270677573579001448798389168], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 3, 2, 1], [2, 4, 1, 1], [2, 6, 1, 1], [2, 6, 4, 1], [2, 12, 1, 1], [2, 12, 2, 1], [2, 24, 2, 1], [3, 1, 2, 1], [3, 128, 1, 1], [3, 128, 2, 1], [4, 12, 2, 1], [4, 24, 1, 1], [4, 24, 2, 1], [4, 24, 4, 1], [4, 48, 4, 1], [6, 3, 2, 1], [6, 3, 4, 1], [6, 4, 2, 1], [6, 6, 2, 1], [6, 6, 8, 1], [6, 12, 2, 1], [6, 12, 4, 1], [6, 24, 4, 1], [6, 128, 1, 1], [6, 128, 2, 1], [12, 12, 4, 1], [12, 24, 2, 1], [12, 24, 4, 1], [12, 24, 8, 1], [12, 48, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6.C_6.C_2^5', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 6376613793978181277, 'autcentquo_nilpotent': False, 'autcentquo_order': 6144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^6.D_6.C_2^3', 'cc_stats': [[1, 1, 1], [2, 3, 3], [2, 4, 1], [2, 6, 5], [2, 12, 3], [2, 24, 2], [3, 1, 2], [3, 128, 3], [4, 12, 2], [4, 24, 7], [4, 48, 4], [6, 3, 6], [6, 4, 2], [6, 6, 10], [6, 12, 6], [6, 24, 4], [6, 128, 3], [12, 12, 4], [12, 24, 14], [12, 48, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '768.1090070', 'commutator_count': 1, 'commutator_label': '192.1541', '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', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 581, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['768.1090070', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 3], [2, 4, 1, 1], [2, 6, 1, 5], [2, 12, 1, 3], [2, 24, 1, 2], [3, 1, 2, 1], [3, 128, 1, 1], [3, 128, 2, 1], [4, 12, 1, 2], [4, 24, 1, 7], [4, 48, 1, 4], [6, 3, 2, 3], [6, 4, 2, 1], [6, 6, 2, 5], [6, 12, 2, 3], [6, 24, 2, 2], [6, 128, 1, 1], [6, 128, 2, 1], [12, 12, 2, 2], [12, 24, 2, 7], [12, 48, 2, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 36, 'exponent': 12, 'exponents_of_order': [8, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 4]], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '576.8670', 'hash': 4285271810845561732, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 2, 2, 2, 2, 2, 1, 3, 2, 2], 'inner_gens': [[93408967995, 40312474, 93405331342, 87091200, 127370880, 16449, 174356582400, 174476734803, 174396888895, 93445593986], [93529081275, 87123994, 93405331342, 40279680, 127370880, 16449, 174356582400, 174436454944, 174443698781, 93492406236], [93408967995, 87123994, 93405331342, 40279680, 127370880, 3758, 174356582400, 174479974086, 174443698781, 93492414105], [93529081275, 87123994, 93405331342, 40279680, 127370880, 3758, 174356582400, 174360590157, 174443698781, 93492414105], [93408967995, 87123994, 93405331342, 40279680, 127370880, 3758, 174356582400, 174404135757, 174443698781, 93492414105], [93408965484, 87096690, 93405331342, 40279680, 127370880, 3758, 174356582400, 174479979231, 174443700415, 93492414105], [93408967995, 87123994, 93405331342, 40279680, 127370880, 3758, 174356582400, 174479977677, 174443698781, 93492414105], [93445234986, 127396061, 93405325075, 87091200, 40279680, 3036, 174356582400, 174479977677, 174483958770, 93532719153], [93529078764, 87123994, 93405331342, 40279680, 127370880, 16449, 174356582400, 174436428486, 174443698781, 93492406236], [93529084804, 87116381, 93405331342, 40279680, 127370880, 3758, 174356582400, 174436416621, 174443706394, 93492414105]], 'inner_hash': 1090070, 'inner_nilpotent': False, 'inner_order': 768, 'inner_split': False, 'inner_tex': 'C_2^5:S_4', 'inner_used': [1, 2, 4, 6, 8], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 12], [2, 6], [3, 36], [6, 30], [12, 6]], 'label': '2304.wi', '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': 'C3*C2^5:S4', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 32, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 90, 'number_divisions': 60, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 838, 'number_subgroup_classes': 2200, 'number_subgroups': 21392, 'old_label': None, 'order': 2304, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 127], [3, 386], [4, 384], [6, 638], [12, 768]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[93408967995, 87118015, 93405331342, 40292755, 127390222, 16449, 174356582400, 174479977677, 174443679090, 93492443231], [93529081275, 87123994, 93405331342, 40279680, 127370880, 127374638, 174356582400, 174479977677, 174443698781, 93532693785], [174480351675, 87123994, 174356601742, 40279680, 127370880, 127374638, 93405312000, 93528707277, 93492428381, 174483964185], [93529092283, 87123994, 93405331342, 40279680, 127370880, 127383221, 174356582400, 174479977677, 174443698781, 93532700765]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [3, 12], [4, 2], [6, 22], [12, 12], [24, 2]], 'representations': {'PC': {'code': '28148781017702955630692430588370491291579089350865480326278719121442826392971936278235820705455424', 'gens': [1, 3, 5, 6, 7, 8, 9, 10], 'pres': [10, -2, -3, -2, -3, -2, 2, 2, 2, 2, 2, 20, 1811, 6302, 642, 82, 6723, 3133, 7204, 1524, 634, 6145, 4175, 2526, 446, 1086, 132487, 26427, 14197, 58328, 9748, 848, 133209, 36029, 7239]}, 'Perm': {'d': 15, 'gens': [93408967995, 87123994, 93405331342, 40279680, 127370880, 3758, 174356582400, 174479977677, 174443698781, 93492414105]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_2^5:S_4', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
