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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1728.47846', 'ambient_counter': 47846, 'ambient_order': 1728, 'ambient_tex': 'C_6^2:\\GL(2,3)', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 4, 'counter': 51, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.47846.24.k1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '24.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': 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': '72.42', 'subgroup_hash': 42, 'subgroup_order': 72, 'subgroup_tex': 'C_3\\times S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.47846', 'aut_centralizer_order': 6, 'aut_label': '24.k1', 'aut_quo_index': None, 'aut_stab_index': 24, 'aut_weyl_group': '24.12', 'aut_weyl_index': 144, 'centralizer': '576.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.b1.a1'], 'contains': ['48.g1.a1', '72.g1.a1', '72.j1.a1', '96.f1.a1'], 'core': '432.a1.a1', 'coset_action_label': None, 'count': 24, 'diagramx': [8861, -1, 700, -1, 331, -1, 273, -1], 'generators': [23, 83630285, 16, 2635961880, 2133815884], 'label': '1728.47846.24.k1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '24.k1.a1', 'old_label': '24.k1.a1', 'projective_image': '1728.47846', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.k1.a1', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', '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, 3, 2, 2], 'aut_gens': [[16247572, 26126314, 34591248, 17540971, 23068812], [28828833, 23068902, 11426365, 700115, 23068812], [25889986, 16366933, 34591248, 28947289, 11534406], [30718424, 347165, 11426365, 23658522, 23068812], [16247572, 23658428, 34591248, 20832407, 23068812], [16247572, 26126314, 34591248, 23658522, 23068812]], 'aut_group': '48.48', 'aut_hash': 48, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 6, 'aut_perms': [403, 316, 181, 314, 82], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 1], [3, 1, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [4, 6, 1, 1], [6, 3, 2, 1], [6, 6, 2, 1], [12, 6, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_4', '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': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 6, 1], [3, 1, 2], [3, 8, 3], [4, 6, 1], [6, 3, 2], [6, 6, 2], [12, 6, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '24.12', 'commutator_count': 1, 'commutator_label': '12.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 42, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['24.12', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 1], [3, 1, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [4, 6, 1, 1], [6, 3, 2, 1], [6, 6, 2, 1], [12, 6, 2, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 36, 'exponent': 12, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '72.42', 'hash': 42, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 2, 3, 1], 'inner_gens': [[16247572, 26126314, 34591248, 23658522, 23068812], [16247572, 26126314, 35544419, 3554814, 23068812], [16247572, 23658428, 34591248, 20832407, 23068812], [28828833, 23068902, 35544419, 17540971, 23068812], [16247572, 26126314, 34591248, 17540971, 23068812]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 3, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 6], [2, 3], [3, 6]], 'label': '72.42', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 4, 'linQ_dim': 5, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3*S4', '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': 10, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 15, 'number_divisions': 10, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 24, 'number_subgroups': 70, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 9], [3, 26], [4, 6], [6, 18], [12, 12]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [5767203], 'outer_gens': [[25889986, 16366933, 34591248, 28947289, 11534406]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [3, 2], [4, 1], [6, 2]], 'representations': {'PC': {'code': 141695920925449, 'gens': [1, 2, 3, 4], 'pres': [5, -2, -3, -2, 2, -3, 41, 322, 363, 548, 58]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [706461086246, 48233675443]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [16247572, 26126314, 34591248, 17540971, 23068812]}, 'Perm': {'d': 7, 'gens': [120, 3, 144, 744, 1680]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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 S_4', 'transitive_degree': 12, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [6, 12, 8], 'aut_gens': [[83630285, 124600323, 138384840, 160524120, 7, 16, 215717040, 689885400, 1090616400], [131364122, 62179332, 2417643367, 2116453816, 16, 23, 3568286160, 689885400, 1540563120], [2094906974, 2555527695, 1727839567, 3433738343, 23, 16, 499092480, 3636230520, 2635961880], [131364122, 279872048, 2600081400, 3153911160, 7, 23, 1571285640, 2055156600, 1540563120]], 'aut_group': None, 'aut_hash': 5785719046187484226, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3456, 'aut_permdeg': 72, 'aut_perms': [46161459384960000495644886075397197123751724746806319332267730795893477944201452615126717470793182088970, 4168897177126923374696390721550719128844996313009807694022184962840616466955477435354192664851673911992, 57189968736850917452475310859516383986763632264973455228654006709652898314931545604064904281722368205004], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [2, 72, 1, 1], [3, 8, 1, 1], [3, 96, 1, 1], [3, 192, 1, 1], [4, 54, 2, 1], [4, 72, 1, 1], [4, 108, 1, 1], [6, 24, 1, 1], [6, 144, 1, 1], [6, 288, 1, 1], [8, 108, 2, 2], [12, 144, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3:S_3.C_2^2:S_4:C_2', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 5785719046187484226, 'autcentquo_nilpotent': False, 'autcentquo_order': 3456, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3:S_3.C_2^2:S_4:C_2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 1], [2, 72, 1], [3, 8, 1], [3, 96, 1], [3, 192, 1], [4, 54, 2], [4, 72, 1], [4, 108, 1], [6, 24, 1], [6, 144, 1], [6, 288, 1], [8, 108, 4], [12, 144, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1728.47846', 'commutator_count': 1, 'commutator_label': '864.4666', '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': 47846, 'cyclic': False, 'derived_length': 5, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [2, 72, 1, 1], [3, 8, 1, 1], [3, 96, 1, 1], [3, 192, 1, 1], [4, 54, 1, 2], [4, 72, 1, 1], [4, 108, 1, 1], [6, 24, 1, 1], [6, 144, 1, 1], [6, 288, 1, 1], [8, 108, 2, 2], [12, 144, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 144, 'exponent': 24, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1728.47846', 'hash': 47846, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 3, 4, 4, 2, 2, 2, 3, 3], 'inner_gens': [[83630285, 244954084, 56287440, 287779680, 23, 16, 215717040, 1540563120, 1090616400], [285038762, 124600323, 160524120, 88277040, 16, 23, 215717040, 2055156600, 1090616400], [211251725, 62179323, 138384840, 56287440, 7, 16, 215717040, 2055156600, 3636230520], [23719805, 4002603, 287779680, 160524120, 7, 16, 215717040, 2635961880, 689885400], [83630294, 124600340, 138384840, 160524120, 7, 16, 215717040, 689885400, 1090616400], [83630285, 124600331, 138384840, 160524120, 7, 16, 215717040, 689885400, 1090616400], [83630285, 124600323, 138384840, 160524120, 7, 16, 215717040, 2882048400, 2635961880], [3891657005, 2436819963, 2445983280, 1965122640, 7, 16, 2937322200, 689885400, 1090616400], [83630285, 124600323, 3837255120, 1644941520, 7, 16, 2682854040, 689885400, 1090616400]], 'inner_hash': 47846, 'inner_nilpotent': False, 'inner_order': 1728, 'inner_split': True, 'inner_tex': 'C_6^2:\\GL(2,3)', 'inner_used': [1, 2, 3, 5, 8], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 2], [2, 3], [3, 6], [4, 1], [6, 3], [8, 2], [16, 1], [24, 2]], 'label': '1728.47846', 'linC_count': 8, 'linC_degree': 11, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 8, 'linQ_dim': 11, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 11, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C6^2:GL(2,3)', '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': 17, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 20, 'number_divisions': 18, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 159, 'number_subgroup_classes': 187, 'number_subgroups': 4746, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 111], [3, 296], [4, 288], [6, 456], [8, 432], [12, 144]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[83630285, 124600323, 138384863, 160524127, 7, 16, 215717040, 689885400, 1090616400]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [3, 6], [4, 2], [6, 1], [8, 2], [12, 1], [16, 1], [24, 2]], 'representations': {'PC': {'code': 13527894247287479705137064282073946012920757372630407245621056627134174811641747327259695076504197253, 'gens': [1, 2, 3, 4, 6, 8], 'pres': [9, 2, 3, 2, 2, 2, 2, 3, 2, 3, 73, 974, 578, 128, 1083, 120, 237, 102, 28526, 7367, 3272, 581, 158, 14136, 6081, 546, 120967, 12112, 4633, 6370, 4363, 214, 108872, 27233, 10394, 683, 3932]}, 'Perm': {'d': 13, 'gens': [83630285, 124600323, 138384840, 160524120, 7, 16, 215717040, 689885400, 1090616400]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2:\\GL(2,3)', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
