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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1008.787', 'ambient_counter': 787, 'ambient_order': 1008, 'ambient_tex': 'D_6.D_{42}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 12, 'counter': 103, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1008.787.42.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '42.f1.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': 42, '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': '24.7', 'subgroup_hash': 7, 'subgroup_order': 24, 'subgroup_tex': 'C_6:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1008.787', 'aut_centralizer_order': 24, 'aut_label': '42.f1', 'aut_quo_index': None, 'aut_stab_index': 21, 'aut_weyl_group': '24.14', 'aut_weyl_index': 504, 'centralizer': '252.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.f1.a1', '14.c1.a1', '21.a1.a1'], 'contains': ['84.b1.a1', '84.n1.a1', '84.o1.a1', '126.c1.a1'], 'core': '84.b1.a1', 'coset_action_label': None, 'count': 21, 'diagramx': [2462, -1, 7083, -1, 3644, -1, 9234, -1], 'generators': [85, 42, 168, 28], 'label': '1008.787.42.f1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '21.a1.a1', 'old_label': '42.f1.a1', 'projective_image': '504.192', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '42.f1.a1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 12], 'aut_gens': [[1, 4], [1, 20], [1, 6], [21, 6]], 'aut_group': '48.38', 'aut_hash': 38, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 7, 'aut_perms': [745, 24, 1707], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 2, 1, 1], [4, 3, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', '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, 2, 1], [4, 3, 4], [6, 2, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '6.1', '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'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.1', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 3, 2, 2], [6, 2, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '12.4', 'hash': 7, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 20], [9, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 4]], 'label': '24.7', 'linC_count': 12, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6:C4', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 12, 'number_divisions': 10, 'number_normal_subgroups': 13, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 22, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 12], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 6], [15, 6]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6]], 'representations': {'PC': {'code': 122930001, 'gens': [1, 3], 'pres': [4, -2, -2, -2, -3, 8, 242, 34, 259]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26129266, 26129427]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [1497706, 72975, 1440589, 1565004]}, 'Perm': {'d': 9, 'gens': [46105, 24, 90720, 3]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6:C_4', '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [6, 6, 6, 6, 6, 6, 6], 'aut_gens': [[1, 2, 84, 336], [41, 218, 420, 336], [177, 38, 84, 336], [17, 202, 924, 336], [241, 202, 84, 336], [225, 242, 84, 672], [233, 214, 252, 336], [185, 218, 588, 672]], 'aut_group': None, 'aut_hash': 6486938177741458836, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12096, 'aut_permdeg': 50, 'aut_perms': [9075657988347242797175946705663955316434890995485205458110922576, 1710269504374676004800697245119696951658996772758548631374395765, 15094130017960256819317170346269828831227396386748200513878755285, 19681672398093571932440970672371986177175125918786895187331580994, 19057856980476496231989216735866278203869958312894807938920305714, 28108208773756790422338412431313709421238933133081990592522870220, 24634567053378230410097069992221256007135512804853549910587712782], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 1], [2, 42, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 6, 1, 1], [4, 42, 1, 1], [4, 63, 2, 1], [4, 126, 1, 1], [6, 2, 1, 2], [6, 4, 1, 3], [6, 4, 2, 1], [6, 12, 1, 1], [6, 84, 1, 1], [7, 2, 3, 1], [12, 12, 1, 1], [12, 84, 1, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 12, 3, 1], [21, 2, 6, 1], [21, 4, 3, 1], [21, 4, 6, 1], [28, 12, 3, 1], [42, 2, 6, 1], [42, 4, 3, 1], [42, 4, 6, 3], [42, 4, 12, 1], [42, 12, 6, 1], [84, 12, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6\\times C_2^2\\times S_3\\times F_7', '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': 42, 'autcentquo_group': '1512.792', 'autcentquo_hash': 792, 'autcentquo_nilpotent': False, 'autcentquo_order': 1512, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 6, 1], [2, 42, 1], [3, 2, 2], [3, 4, 1], [4, 6, 1], [4, 42, 1], [4, 63, 2], [4, 126, 1], [6, 2, 2], [6, 4, 5], [6, 12, 1], [6, 84, 1], [7, 2, 3], [12, 12, 1], [12, 84, 1], [14, 2, 3], [14, 4, 3], [14, 12, 3], [21, 2, 6], [21, 4, 9], [28, 12, 3], [42, 2, 6], [42, 4, 33], [42, 12, 6], [84, 12, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '504.192', 'commutator_count': 1, 'commutator_label': '126.16', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 787, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 1], [2, 42, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 6, 1, 1], [4, 42, 1, 1], [4, 63, 2, 1], [4, 126, 1, 1], [6, 2, 1, 2], [6, 4, 1, 5], [6, 12, 1, 1], [6, 84, 1, 1], [7, 2, 3, 1], [12, 12, 1, 1], [12, 84, 1, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 12, 3, 1], [21, 2, 6, 1], [21, 4, 3, 1], [21, 4, 6, 1], [28, 12, 3, 1], [42, 2, 6, 1], [42, 4, 3, 1], [42, 4, 6, 5], [42, 12, 6, 1], [84, 12, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 21504, 'exponent': 84, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[4, -1, 12]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '504.192', 'hash': 787, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 42, 2, 3], 'inner_gens': [[1, 250, 84, 336], [173, 2, 252, 336], [1, 170, 84, 672], [1, 2, 756, 336]], 'inner_hash': 192, 'inner_nilpotent': False, 'inner_order': 504, 'inner_split': True, 'inner_tex': 'S_3\\times D_{42}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 24, 'irrQ_dim': 48, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 46], [4, 51]], 'label': '1008.787', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 8, 'linQ_dim': 14, 'linQ_dim_count': 80, 'linR_count': 120, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D6.D42', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 38, 'number_characteristic_subgroups': 50, 'number_conjugacy_classes': 105, 'number_divisions': 40, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 174, 'number_subgroup_classes': 176, 'number_subgroups': 1628, 'old_label': None, 'order': 1008, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 51], [3, 8], [4, 300], [6, 120], [7, 6], [12, 96], [14, 54], [21, 48], [28, 36], [42, 216], [84, 72]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 58, 84, 336], [1, 58, 84, 672], [1, 38, 252, 672]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [744, 24, 40323], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 7], [6, 4], [12, 7], [24, 6]], 'representations': {'PC': {'code': 39848191768010354821140597467226697, 'gens': [1, 2, 5, 7], 'pres': [7, -2, -2, -3, -7, -2, -2, -3, 3501, 36, 1682, 79, 2019, 4421, 102, 426]}, 'Perm': {'d': 21, 'gens': [2446776433221161065, 5245030977965856000, 5353870023248947200, 7939008778821120000, 518918400, 3, 444984]}}, 'schur_multiplier': [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_{42}', 'transitive_degree': 168, 'wreath_data': None, 'wreath_product': False}
