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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '11664.bi', 'ambient_counter': 35, 'ambient_order': 11664, 'ambient_tex': 'C_3^5:(C_2\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 1, 'counter': 252, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '11664.bi.81.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '81.d1', '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': 81, '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': '144.153', 'subgroup_hash': 153, 'subgroup_order': 144, 'subgroup_tex': 'D_6:D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '11664.bi', 'aut_centralizer_order': None, 'aut_label': '81.d1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '5832.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['9.c1', '27.e1'], 'contains': ['162.f1', '162.h1', '162.m1', '162.t1', '162.v1', '162.bf1', '162.bg1', '243.b1', '243.d1'], 'core': '11664.a1', 'coset_action_label': None, 'count': 81, 'diagramx': [9528, -1, 9260, -1], 'generators': [376610297862964, 355694133899279, 3778153297350210, 3778897055109384, 2623482797886479, 93405312144], 'label': '11664.bi.81.d1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '81.d1', 'old_label': '81.d1', 'projective_image': '11664.bi', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '81.d1', 'subgroup_fusion': None, 'weyl_group': '72.46'}
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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': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 2, 2, 6, 6], 'aut_gens': [[1, 85704, 367920, 806400, 144, 3], [1, 85704, 367920, 806400, 144, 4], [1, 766104, 367920, 806400, 144, 3], [1, 85704, 1174320, 806400, 240, 3], [5, 85704, 1174320, 806400, 144, 3], [806401, 85800, 367920, 806400, 144, 3]], 'aut_group': '288.1040', 'aut_hash': 1040, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 12, 'aut_perms': [90726, 41040, 3760560, 90751, 79924320], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 2, 1], [2, 6, 1, 2], [2, 18, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 6, 1, 1], [4, 18, 1, 1], [6, 2, 1, 2], [6, 2, 2, 1], [6, 4, 1, 2], [6, 4, 2, 1], [6, 6, 2, 2], [6, 12, 1, 1], [12, 12, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_6^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 3, 2], [2, 6, 2], [2, 18, 1], [3, 2, 2], [3, 4, 1], [4, 6, 1], [4, 18, 1], [6, 2, 4], [6, 4, 4], [6, 6, 4], [6, 12, 1], [12, 12, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '72.46', 'commutator_count': 1, 'commutator_label': '18.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 153, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['24.8', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 6, 1, 2], [2, 18, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 6, 1, 1], [4, 18, 1, 1], [6, 2, 1, 2], [6, 2, 2, 1], [6, 4, 1, 2], [6, 4, 2, 1], [6, 6, 1, 2], [6, 6, 2, 1], [6, 12, 1, 1], [12, 12, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2688, 'exponent': 12, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '72.46', 'hash': 153, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 2, 1, 3, 3], 'inner_gens': [[1, 85704, 367920, 806400, 144, 4], [1, 85704, 1174320, 806400, 240, 3], [1, 766104, 367920, 806400, 144, 3], [1, 85704, 367920, 806400, 144, 3], [1, 85944, 367920, 806400, 144, 3], [5, 85704, 367920, 806400, 144, 3]], 'inner_hash': 46, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'S_3\\times D_6', 'inner_used': [1, 2, 3, 5, 6], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 14], [4, 5]], 'label': '144.153', 'linC_count': 18, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 48, 'linQ_dim': 6, 'linQ_dim_count': 48, 'linR_count': 48, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D6:D6', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 22, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 27, 'number_divisions': 24, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 104, 'number_subgroup_classes': 116, 'number_subgroups': 412, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 39], [3, 8], [4, 24], [6, 60], [12, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[806401, 85704, 367920, 806400, 240, 4], [1, 85704, 367920, 806400, 240, 3]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 5], [8, 1]], 'representations': {'PC': {'code': 4214139732408153349, 'gens': [1, 2, 3, 5], 'pres': [6, -2, -2, -2, -3, -2, -3, 1658, 50, 387, 1810, 88, 1739]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125124615946038433, 125101729074820244, 58301502094248184]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [19949, 18685, 18694, 1777, 1733, 9587]}, 'Perm': {'d': 10, 'gens': [1, 85704, 367920, 806400, 144, 3]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_6:D_6', 'transitive_degree': 24, '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': 4, 'aut_exponent': 36, 'aut_gen_orders': [6, 12, 2], 'aut_gens': [[3422471836412425, 3000455699834404, 1845866591205387], [733236182035205, 6024636436901283, 3779091441498509], [733142820233765, 1866420289906532, 1510927639925160], [3422552824015946, 6025307394712469, 1845785719813704]], 'aut_group': '23328.p', 'aut_hash': 1081498020297448794, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23328, 'aut_permdeg': 63, 'aut_perms': [1079286075495160614108336252798224612490625801215013281727386392803559190022049770503283, 1612756608600375176106900168120880004770260113591436142016315303036150276841120974483494, 1109681937880568982896520705299963758900508586329172232472360408660003143942212925917656], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [2, 54, 1, 2], [2, 243, 1, 1], [3, 2, 1, 1], [3, 4, 2, 1], [3, 6, 1, 2], [3, 8, 2, 1], [3, 12, 1, 3], [3, 24, 1, 3], [3, 24, 2, 1], [3, 48, 1, 1], [3, 216, 1, 1], [3, 432, 1, 1], [4, 486, 1, 2], [6, 36, 2, 1], [6, 54, 1, 4], [6, 54, 2, 2], [6, 108, 1, 8], [6, 108, 2, 4], [6, 216, 1, 3], [6, 216, 2, 2], [6, 486, 1, 1], [6, 648, 1, 1], [9, 216, 2, 1], [9, 432, 2, 1], [12, 972, 1, 2], [18, 648, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_3^2\\times S_3^3):D_6', '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': 36, 'autcentquo_group': '23328.p', 'autcentquo_hash': 1081498020297448794, 'autcentquo_nilpotent': False, 'autcentquo_order': 23328, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(C_3^2\\times S_3^3):D_6', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 27, 1], [2, 54, 2], [2, 243, 1], [3, 2, 1], [3, 4, 2], [3, 6, 2], [3, 8, 2], [3, 12, 3], [3, 24, 5], [3, 48, 1], [3, 216, 1], [3, 432, 1], [4, 486, 2], [6, 36, 2], [6, 54, 8], [6, 108, 16], [6, 216, 7], [6, 486, 1], [6, 648, 1], [9, 216, 2], [9, 432, 2], [12, 972, 2], [18, 648, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '11664.bi', 'commutator_count': 1, 'commutator_label': '2916.ei', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 35, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [2, 54, 1, 2], [2, 243, 1, 1], [3, 2, 1, 1], [3, 4, 2, 1], [3, 6, 1, 2], [3, 8, 2, 1], [3, 12, 1, 3], [3, 24, 1, 3], [3, 24, 2, 1], [3, 48, 1, 1], [3, 216, 1, 1], [3, 432, 1, 1], [4, 486, 1, 2], [6, 36, 2, 1], [6, 54, 1, 4], [6, 54, 2, 2], [6, 108, 1, 8], [6, 108, 2, 4], [6, 216, 1, 3], [6, 216, 2, 2], [6, 486, 1, 1], [6, 648, 1, 1], [9, 216, 2, 1], [9, 432, 2, 1], [12, 972, 1, 2], [18, 648, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 702, 'exponent': 36, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 2], [12, 1, 2], [24, 0, 4], [24, 1, 4], [48, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '3888.cb', 'hash': 7326314448994360356, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 6, 12], 'inner_gens': [[3422471836412425, 6024462080354160, 5267487106108558], [21010447042412, 3000455699834404, 2645049972937434], [733317176851441, 1155890297237042, 1845866591205387]], 'inner_hash': 7326314448994360356, 'inner_nilpotent': False, 'inner_order': 11664, 'inner_split': True, 'inner_tex': 'C_3^5:(C_2\\times S_4)', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 4], [3, 4], [4, 9], [6, 14], [8, 8], [12, 13], [16, 2], [24, 10], [48, 1]], 'label': '11664.bi', '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': 'C3^5:(C2*S4)', 'ngens': 3, '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, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 54, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 69, 'number_divisions': 54, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 1146, 'number_subgroup_classes': 1151, 'number_subgroups': 76244, 'old_label': None, 'order': 11664, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 387], [3, 890], [4, 972], [6, 4878], [9, 1296], [12, 1944], [18, 1296]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3757410831167401, 3000549025312704, 1845785683480344]], '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': 7, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 10], [8, 4], [12, 11], [16, 4], [24, 8], [32, 1], [48, 3]], 'representations': {'PC': {'code': '1868343611568901975010497460661645328431399788828267815623933068650673368003070015500072165633085591793652492195308504612814821592156518005935', 'gens': [1, 2, 4, 6, 8, 9, 10], 'pres': [10, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 10800, 114441, 51, 63962, 54022, 379683, 48013, 13823, 113, 45604, 12014, 5724, 47525, 137175, 125845, 55835, 175, 80646, 211696, 75626, 6756, 34577, 816488, 207378, 255988, 48638, 3298, 1036809, 50419, 129629, 10859]}, 'Perm': {'d': 18, 'gens': [3422471836412425, 3000455699834404, 1845866591205387]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^5:(C_2\\times S_4)', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
