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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '24192.u', 'ambient_counter': 21, 'ambient_order': 24192, 'ambient_tex': 'D_4\\times \\SL(2,8):C_6', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 8, 'counter': 69, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '24192.u.112.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '112.f1', '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': 112, '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': '216.75', 'subgroup_hash': 75, 'subgroup_order': 216, 'subgroup_tex': 'C_{18}:C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '24192.u', 'aut_centralizer_order': None, 'aut_label': '112.f1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1008.l1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['56.c1', '56.d1', '56.e1'], 'contains': ['224.e1', '224.j1', '336.l1', '336.n1', '336.o1'], 'core': '3024.b1', 'coset_action_label': None, 'count': 28, 'diagramx': [9153, -1, 465, -1], 'generators': [87178398545, 87178317323, 87178291200, 87178465239, 1520467200, 961632000], 'label': '24192.u.112.f1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '28.a1', 'old_label': '112.f1', 'projective_image': '6048.r', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '112.f1', 'subgroup_fusion': None, 'weyl_group': '36.12'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '72.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 6, 6, 6], 'aut_gens': [[1, 6], [145, 214], [1, 151], [37, 80], [1, 42]], 'aut_group': '432.390', 'aut_hash': 390, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 432, 'aut_permdeg': 13, 'aut_perms': [11330040, 4036763527, 18518405, 566225296], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 2, 1], [3, 3, 1, 2], [4, 1, 4, 1], [6, 1, 2, 1], [6, 1, 4, 1], [6, 3, 1, 2], [6, 3, 2, 2], [9, 3, 6, 1], [12, 1, 8, 1], [12, 3, 4, 2], [18, 3, 6, 1], [18, 3, 12, 1], [36, 3, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6^2:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '72.37', 'autcent_hash': 37, 'autcent_nilpotent': True, 'autcent_order': 72, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4\\times C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 1, 2], [3, 3, 2], [4, 1, 4], [6, 1, 6], [6, 3, 6], [9, 3, 6], [12, 1, 8], [12, 3, 8], [18, 3, 18], [36, 3, 24]], 'center_label': '24.9', 'center_order': 24, 'central_product': True, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 75, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['27.4', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [3, 3, 2, 1], [4, 1, 2, 2], [6, 1, 2, 3], [6, 3, 2, 3], [9, 3, 2, 3], [12, 1, 4, 2], [12, 3, 4, 2], [18, 3, 2, 9], [36, 3, 4, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 36, 'exponents_of_order': [3, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '36.14', 'hash': 75, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3], 'inner_gens': [[1, 150], [73, 6]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': True, 'inner_tex': 'C_3^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 72], [3, 16]], 'label': '216.75', 'linC_count': 432, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C18:C12', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 20, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 88, 'number_divisions': 36, 'number_normal_subgroups': 56, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 64, 'number_subgroups': 80, 'old_label': None, 'order': 216, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 3], [3, 8], [4, 4], [6, 24], [9, 18], [12, 32], [18, 54], [36, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [2, 0, 0], 'outer_gens': [[1, 174], [1, 9], [109, 119]], 'outer_group': '48.38', 'outer_hash': 38, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 7, 'outer_perms': [1, 744, 2163], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 8], [6, 4], [12, 2]], 'representations': {'PC': {'code': 69322736588186984520993961811, 'gens': [1, 3], 'pres': [6, -2, -3, -2, -2, -3, -3, 12, 2702, 710, 50, 2019, 1881, 69, 5044, 1450, 118]}, 'GLZN': {'d': 2, 'p': 36, 'gens': [1166413, 47305, 1656647, 47089, 886483, 638377]}, 'Perm': {'d': 15, 'gens': [1516838400, 87178291200, 357120, 243, 482630400, 80884]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 12], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{18}:C_{12}', 'transitive_degree': 72, '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 252, 'aut_gen_orders': [18, 6, 36, 6], 'aut_gens': [[1037836800, 1520467200, 87178291200, 88140202024, 39916800, 87218432753], [1037836800, 1520467200, 88698758400, 88140220133, 1480550400, 87218341004], [1037836800, 1520467200, 87178291200, 87737197906, 1480550400, 88659086269], [39916800, 1520467200, 87178291200, 558921674, 88216128000, 1038122870], [88216128000, 1520467200, 87178291200, 559165045, 1480550400, 88658887293]], 'aut_group': None, 'aut_hash': 7859322611675049309, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96768, 'aut_permdeg': 92, 'aut_perms': [456473527738953639051728649077445307842070617748090041986590954250771391037694814142696890044950524418036338293410558068154179488988140008268, 3510030158524706444330836333240312442853203726160650334555259146205916725243807347868882461209101294622824308657568403311312877013072451713936, 6511625021717981358312711868607524042794742507516472166400620754169464644692526166040368991853822275135900638556122872782231573378230662563985, 5316059230449708415666229564653937064272063960545339690783224581580225153646468850276236361003789974273581693324559010932329768576250778548743], 'aut_phi_ratio': 14.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [2, 63, 1, 2], [2, 63, 2, 1], [2, 126, 4, 1], [3, 56, 1, 1], [3, 84, 1, 2], [4, 2, 2, 1], [4, 126, 2, 1], [6, 56, 1, 1], [6, 56, 2, 1], [6, 84, 1, 2], [6, 84, 2, 2], [6, 112, 4, 1], [6, 168, 4, 2], [6, 252, 1, 4], [6, 252, 2, 2], [6, 504, 4, 2], [7, 216, 1, 1], [9, 168, 1, 3], [12, 112, 2, 1], [12, 168, 2, 2], [12, 504, 2, 2], [14, 216, 1, 1], [14, 216, 2, 1], [14, 432, 4, 1], [18, 168, 1, 3], [18, 168, 2, 3], [18, 336, 4, 3], [28, 432, 2, 1], [36, 336, 2, 3]], 'aut_supersolvable': False, 'aut_tex': '\\SL(2,8).C_3\\times C_2\\wr C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 126, 'autcentquo_group': '3024.r', 'autcentquo_hash': 5679810977767407995, 'autcentquo_nilpotent': False, 'autcentquo_order': 3024, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SL(2,8):C_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [2, 63, 4], [2, 126, 4], [3, 56, 1], [3, 84, 2], [4, 2, 2], [4, 126, 2], [6, 56, 3], [6, 84, 6], [6, 112, 4], [6, 168, 8], [6, 252, 8], [6, 504, 8], [7, 216, 1], [9, 168, 3], [12, 112, 2], [12, 168, 4], [12, 504, 4], [14, 216, 3], [14, 432, 4], [18, 168, 9], [18, 336, 12], [28, 432, 2], [36, 336, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '6048.r', 'commutator_count': 1, 'commutator_label': '1008.880', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '504.156'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 21, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1512.779', 1], ['2.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [2, 63, 1, 4], [2, 126, 1, 4], [3, 56, 1, 1], [3, 84, 2, 1], [4, 2, 1, 2], [4, 126, 1, 2], [6, 56, 1, 3], [6, 84, 2, 3], [6, 112, 1, 4], [6, 168, 2, 4], [6, 252, 2, 4], [6, 504, 2, 4], [7, 216, 1, 1], [9, 168, 1, 1], [9, 168, 2, 1], [12, 112, 1, 2], [12, 168, 2, 2], [12, 504, 2, 2], [14, 216, 1, 3], [14, 432, 1, 4], [18, 168, 1, 3], [18, 168, 2, 3], [18, 336, 1, 4], [18, 336, 2, 4], [28, 432, 1, 2], [36, 336, 1, 2], [36, 336, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 45575712, 'exponent': 252, 'exponents_of_order': [7, 3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '12096.br', 'hash': 38981159678273474, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 126, 'inner_gen_orders': [2, 1, 1, 6, 2, 6], 'inner_gens': [[1037836800, 1520467200, 87178291200, 87737405224, 1480550400, 88659066353], [1037836800, 1520467200, 87178291200, 88140202024, 39916800, 87218432753], [1037836800, 1520467200, 87178291200, 88140202024, 39916800, 87218432753], [482630400, 1520467200, 87178291200, 88140202024, 1480550400, 88658951698], [482630400, 1520467200, 87178291200, 87737405224, 39916800, 87218432753], [482630400, 1520467200, 87178291200, 87737290094, 39916800, 87218432753]], 'inner_hash': 246442942980519397, 'inner_nilpotent': False, 'inner_order': 6048, 'inner_split': False, 'inner_tex': 'C_2^2\\times {}^2G(2,3)', 'inner_used': [1, 4, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 6], [7, 24], [8, 24], [14, 6], [16, 6], [21, 8], [27, 8], [42, 2], [54, 2]], 'label': '24192.u', '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': False, 'name': 'D4*SL(2,8):C6', 'ngens': 6, '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, 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': 55, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 110, 'number_divisions': 80, 'number_normal_subgroups': 57, 'number_subgroup_autclasses': 544, 'number_subgroup_classes': 1503, 'number_subgroups': 155898, 'old_label': None, 'order': 24192, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 767], [3, 224], [4, 256], [6, 8512], [7, 216], [9, 504], [12, 2912], [14, 2376], [18, 5544], [28, 864], [36, 2016]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 558835200, 0], 'outer_gens': [[39916800, 1520467200, 87178291200, 87737405224, 1037836800, 88216352753], [88216128000, 1520467200, 87178291200, 961910824, 39916800, 87218432753], [88216128000, 1520467200, 88698758400, 87737405224, 87218208000, 1480775153], [88216128000, 1520467200, 87178291200, 88140202024, 87218208000, 40141553]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [288, 121, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 2], [7, 8], [8, 8], [14, 10], [16, 10], [21, 8], [27, 8], [28, 2], [32, 2], [42, 2], [54, 2]], 'representations': {'Perm': {'d': 15, 'gens': [1037836800, 1520467200, 87178291200, 88140202024, 39916800, 87218432753]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_4\\times \\SL(2,8):C_6', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
