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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '224.74', 'ambient_counter': 74, 'ambient_order': 224, 'ambient_tex': 'C_2^3.D_{14}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 16, 'counter': 16, 'cyclic': False, 'direct': None, 'hall': 2, 'label': '224.74.7.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '7.a1.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': 7, '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': '32.33', 'subgroup_hash': 33, 'subgroup_order': 32, 'subgroup_tex': 'C_4^2:C_2', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '224.74', 'aut_centralizer_order': 6, 'aut_label': '7.a1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '64.267', 'aut_weyl_index': 42, 'centralizer': '56.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['14.a1.a1', '14.b1.a1', '14.c1.a1', '14.d1.a1', '14.e1.a1', '14.f1.a1', '14.g1.a1'], 'core': '14.a1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [8284, -1, 8770, -1, 3778, -1, 8031, -1], 'generators': [1, 2, 56], 'label': '224.74.7.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '7.a1.a1', 'projective_image': '56.12', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '7.a1.a1', 'subgroup_fusion': None, 'weyl_group': '8.5'}
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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, 3, 2, 2, 2, 2], 'aut_gens': [[1, 2, 8], [1, 22, 12], [1, 6, 24], [1, 26, 22], [17, 22, 24], [5, 6, 28], [17, 2, 8], [5, 2, 8]], 'aut_group': '192.1540', 'aut_hash': 1540, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 12, 'aut_perms': [143733964, 94586093, 314298665, 94560744, 14953955, 345943829, 98212372], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 4, 1, 1], [4, 2, 6, 1], [4, 4, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\wr C_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 3, 'autcentquo_group': '3.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 3, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 1], [4, 2, 6], [4, 4, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 33, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [4, 2, 2, 3], [4, 4, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 56, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '8.5', 'hash': 33, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[1, 22, 12], [21, 2, 8], [5, 2, 8]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': True, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 6]], 'label': '32.33', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 3, 'linQ_dim': 8, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2:C2', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 14, 'number_divisions': 11, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 30, 'number_subgroups': 42, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3, 2, 2], 'outer_gen_pows': [0, 1, 0, 0], 'outer_gens': [[1, 6, 24], [1, 10, 2], [1, 2, 28], [1, 6, 28]], 'outer_group': '24.13', 'outer_hash': 13, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 6, 'outer_perms': [316, 181, 269, 314], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times A_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 3]], 'representations': {'PC': {'code': 2231959642, 'gens': [1, 2, 4], 'pres': [5, 2, 2, 2, 2, 2, 221, 26, 243, 58]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 30594431879, 40849247818, 30704310001, 61245322049]}, 'Perm': {'d': 16, 'gens': [40291223, 15316555121280, 8395940079936, 2789792421136, 1313941673647]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2:C_2', 'transitive_degree': 16, '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, 14, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 8], [1, 2, 136], [5, 162, 8], [113, 2, 8], [1, 6, 8], [1, 114, 8], [1, 2, 12], [1, 2, 120]], 'aut_group': '2688.ei', 'aut_hash': 1349820288041980446, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2688, 'aut_permdeg': 36, 'aut_perms': [186466113241385276474410801375272832890521, 112570557013751951763176013903830064542068, 3551748755540714423664329420623389260304, 83214627753052840767547577863015373848817, 238056341695658193173648705343678986453657, 238736731789629903848398888432383510, 181176861210384444124445485011348630], 'aut_phi_ratio': 28.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [4, 2, 2, 1], [4, 4, 1, 1], [4, 14, 2, 2], [4, 28, 1, 2], [7, 2, 3, 1], [14, 2, 3, 3], [14, 4, 6, 1], [28, 4, 6, 2]], 'aut_supersolvable': True, 'aut_tex': 'F_7\\times C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 1], [4, 2, 2], [4, 4, 1], [4, 14, 4], [4, 28, 2], [7, 2, 3], [14, 2, 9], [14, 4, 6], [28, 4, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '56.12', 'commutator_count': 1, 'commutator_label': '28.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 74, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [4, 2, 2, 1], [4, 4, 1, 1], [4, 14, 2, 2], [4, 28, 1, 2], [7, 2, 3, 1], [14, 2, 3, 3], [14, 4, 6, 1], [28, 4, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 28, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '56.12', 'hash': 74, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 2, 14], 'inner_gens': [[1, 114, 12], [113, 2, 216], [5, 18, 8]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': True, 'inner_tex': 'C_2\\times D_{14}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 30], [4, 6]], 'label': '224.74', 'linC_count': 48, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 12, 'linQ_dim': 14, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.D14', 'ngens': 6, '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': 18, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 44, 'number_divisions': 18, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 60, 'number_subgroups': 198, 'old_label': None, 'order': 224, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 120], [7, 6], [14, 42], [28, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[5, 2, 8], [113, 2, 8], [1, 6, 8], [1, 2, 40]], 'outer_group': '48.52', 'outer_hash': 52, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [3628800, 744, 40320, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 3], [6, 4], [12, 2], [24, 1]], 'representations': {'PC': {'code': 6959155091463047561309481605, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -2, 2, -2, -7, 1369, 31, 291, 2601, 69, 3130, 88, 3467]}, 'Perm': {'d': 23, 'gens': [1285085920066109683327, 2458122556735945209600, 272743613775864766080, 3698058463769276772480, 4871645936519381550720, 973]}}, 'schur_multiplier': [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': 'C_2^3.D_{14}', 'transitive_degree': 112, 'wreath_data': None, 'wreath_product': False}
