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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '7776.gl', 'ambient_counter': 168, 'ambient_order': 7776, 'ambient_tex': 'C_6^2.S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 9, 'counter': 1456, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '7776.gl.216.dx1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '216.dx1', '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': 216, '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': '36.13', 'subgroup_hash': 13, 'subgroup_order': 36, 'subgroup_tex': 'C_6:S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7776.gl', 'aut_centralizer_order': None, 'aut_label': '216.dx1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1944.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['72.ci1', '72.cj1', '72.ck1', '108.bi1', '108.cs1', '108.dw1', '108.fw1', '108.gl1', '108.hw1', '108.ke1'], 'contains': ['432.b1', '432.et1', '648.ch1', '648.ci1', '648.cj1'], 'core': '864.a1', 'coset_action_label': None, 'count': 27, 'diagramx': None, 'generators': [9597399840000, 1680, 355499061865, 4], 'label': '7776.gl.216.dx1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.i1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.e1', 'old_label': '216.dx1', 'projective_image': '7776.gl', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '216.dx1', 'subgroup_fusion': None, 'weyl_group': '72.46'}
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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': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 4, 2, 4, 3, 2, 6, 6], 'aut_gens': [[1, 2, 6], [19, 2, 6], [35, 26, 32], [25, 4, 30], [33, 14, 34], [5, 2, 6], [19, 28, 6], [33, 2, 6], [23, 16, 30]], 'aut_group': '864.4661', 'aut_hash': 4661, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 11, 'aut_perms': [1, 10889689, 24054000, 20507935, 14605950, 2245249, 19496545, 16180705], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [3, 2, 4, 1], [6, 2, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_3^2:\\GL(2,3)', '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': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 2, 4], [6, 2, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 2, 1, 4], [6, 2, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 21, 'exponent': 6, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '36.13', 'hash': 13, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[1, 4, 30], [5, 2, 6], [13, 2, 6]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 8]], 'label': '36.13', 'linC_count': 18, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 18, 'linQ_dim': 4, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6:S3', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 12, 'number_divisions': 12, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 30, 'number_subgroups': 78, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 19], [3, 8], [6, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 1, 1], 'outer_gens': [[1, 24, 20], [19, 2, 6], [1, 24, 34], [1, 28, 8], [1, 16, 10]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [143, 127, 576, 126, 121], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8]], 'representations': {'PC': {'code': 405218245, 'gens': [1, 2, 3], 'pres': [4, -2, -3, -2, -3, 33, 362, 34, 387]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [12016555, 35931151, 11780031]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8156, 8101, 13286]}, 'Perm': {'d': 8, 'gens': [721, 24, 5763, 5760]}}, '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_6:S_3', 'transitive_degree': 18, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [18, 9, 12, 6], 'aut_gens': [[107336275945, 274032460923, 1496408983345], [349232085629, 8290243987468, 6913917079349], [9354546031225, 8045431351323, 1757940226801], [10905513253225, 4280042593563, 8458221180505], [10556763800426, 5324226930844, 5263280467346]], 'aut_group': None, 'aut_hash': 2074683354935437783, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23328, 'aut_permdeg': 48, 'aut_perms': [3338022680984135420321287486452335142855872418666762422458794, 8386463913282255331929655360164122711133915635379393700133269, 6575949640407834772235204882894819719456725743592250507004322, 5230236811897961408947061979972147239582238833079108801891817], 'aut_phi_ratio': 9.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 2], [2, 9, 1, 1], [2, 18, 1, 1], [2, 27, 1, 1], [2, 54, 1, 2], [2, 81, 1, 2], [2, 162, 1, 1], [2, 243, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 1], [3, 6, 3, 1], [3, 12, 1, 1], [3, 12, 3, 1], [3, 72, 1, 1], [3, 144, 1, 1], [4, 18, 1, 1], [4, 54, 1, 2], [4, 162, 1, 1], [6, 6, 1, 4], [6, 6, 3, 1], [6, 12, 1, 3], [6, 12, 3, 2], [6, 18, 1, 3], [6, 18, 3, 2], [6, 24, 1, 1], [6, 24, 3, 1], [6, 36, 1, 3], [6, 36, 3, 2], [6, 54, 1, 1], [6, 72, 1, 1], [6, 72, 3, 1], [6, 108, 1, 3], [6, 108, 3, 1], [6, 162, 1, 1], [6, 216, 1, 3], [6, 324, 1, 1], [6, 432, 1, 1], [6, 648, 1, 1], [9, 144, 1, 1], [9, 288, 1, 1], [12, 36, 1, 2], [12, 36, 3, 1], [12, 72, 1, 1], [12, 72, 3, 1], [12, 108, 1, 3], [12, 108, 3, 1], [12, 216, 1, 1], [12, 324, 1, 1], [18, 432, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.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': 36, 'autcentquo_group': None, 'autcentquo_hash': 2074683354935437783, 'autcentquo_nilpotent': False, 'autcentquo_order': 23328, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.C_3^4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 3, 2], [2, 9, 1], [2, 18, 1], [2, 27, 1], [2, 54, 2], [2, 81, 2], [2, 162, 1], [2, 243, 1], [3, 2, 2], [3, 4, 1], [3, 6, 4], [3, 12, 4], [3, 72, 1], [3, 144, 1], [4, 18, 1], [4, 54, 2], [4, 162, 1], [6, 6, 7], [6, 12, 9], [6, 18, 9], [6, 24, 4], [6, 36, 9], [6, 54, 1], [6, 72, 4], [6, 108, 6], [6, 162, 1], [6, 216, 3], [6, 324, 1], [6, 432, 1], [6, 648, 1], [9, 144, 1], [9, 288, 1], [12, 36, 5], [12, 72, 4], [12, 108, 6], [12, 216, 1], [12, 324, 1], [18, 432, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '7776.gl', 'commutator_count': 1, 'commutator_label': '972.590', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 168, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['1296.1790', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 2], [2, 9, 1, 1], [2, 18, 1, 1], [2, 27, 1, 1], [2, 54, 1, 2], [2, 81, 1, 2], [2, 162, 1, 1], [2, 243, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 4], [3, 12, 1, 4], [3, 72, 1, 1], [3, 144, 1, 1], [4, 18, 1, 1], [4, 54, 1, 2], [4, 162, 1, 1], [6, 6, 1, 7], [6, 12, 1, 9], [6, 18, 1, 9], [6, 24, 1, 4], [6, 36, 1, 9], [6, 54, 1, 1], [6, 72, 1, 4], [6, 108, 1, 6], [6, 162, 1, 1], [6, 216, 1, 3], [6, 324, 1, 1], [6, 432, 1, 1], [6, 648, 1, 1], [9, 144, 1, 1], [9, 288, 1, 1], [12, 36, 1, 5], [12, 72, 1, 4], [12, 108, 1, 6], [12, 216, 1, 1], [12, 324, 1, 1], [18, 432, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 4354560, 'exponent': 36, 'exponents_of_order': [5, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 6], [24, 1, 3]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '864.4673', 'hash': 1780735244374693297, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 6, 6], 'inner_gens': [[107336275945, 274032463108, 2901955755121], [107336276885, 274032460923, 5793085268189], [9248211304825, 7939579254028, 1496408983345]], 'inner_hash': 1780735244374693297, 'inner_nilpotent': False, 'inner_order': 7776, 'inner_split': True, 'inner_tex': 'C_6^2.S_3^3', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 12], [3, 8], [4, 6], [6, 40], [8, 1], [12, 26], [24, 4]], 'label': '7776.gl', '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': 'C6^2.S3^3', '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, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 75, 'number_characteristic_subgroups': 60, 'number_conjugacy_classes': 105, 'number_divisions': 105, 'number_normal_subgroups': 60, 'number_subgroup_autclasses': 2446, 'number_subgroup_classes': 3692, 'number_subgroups': 153298, 'old_label': None, 'order': 7776, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 735], [3, 296], [4, 288], [6, 3936], [9, 432], [12, 1656], [18, 432]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [0], 'outer_gens': [[9247648839145, 9703701907323, 1496408983345]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 6], [6, 40], [8, 1], [12, 26], [24, 4]], 'representations': {'PC': {'code': '1145904493089241188259747712817117703154240063630190822819729958185754227087453093309537560749896537766763003948205', 'gens': [1, 2, 4, 6, 8, 9], 'pres': [10, 2, 2, 3, 2, 3, 2, 3, 3, 2, 3, 53280, 154761, 51, 34082, 215523, 110173, 9263, 113, 24004, 151214, 6024, 237605, 120975, 59425, 19115, 175, 10096, 34476, 243857, 70107, 69138, 83188, 268, 132019, 141629]}, 'Perm': {'d': 16, 'gens': [107336275945, 274032460923, 1496408983345]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2.S_3^3', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
