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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1152.155870', 'ambient_counter': 155870, 'ambient_order': 1152, 'ambient_tex': 'C_3:\\OD_{16}\\times S_4', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 1, 'counter': 586, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1152.155870.288.o1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '288.o1.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': 288, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1152.155870', 'aut_centralizer_order': 192, 'aut_label': '288.o1', 'aut_quo_index': None, 'aut_stab_index': 24, 'aut_weyl_group': '1.1', 'aut_weyl_index': 4608, 'centralizer': '12.j1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['96.z1.a1', '96.ba1.a1', '144.d1.a1', '144.bl1.a1', '144.bo1.a1'], 'contains': ['576.b1.a1', '576.f1.a1', '576.g1.a1'], 'core': '1152.a1.a1', 'coset_action_label': None, 'count': 12, 'diagramx': [1616, -1, 8419, -1, 1342, -1, 1576, -1], 'generators': [1, 2], 'label': '1152.155870.288.o1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '12.j1.a1', 'old_label': '288.o1.a1', 'projective_image': '1152.155870', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '288.o1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 2, 3, 4, 2, 12, 2, 3, 12, 2, 2], 'aut_gens': [[1, 2, 4, 96, 192], [49, 594, 268, 672, 864], [1, 50, 4, 96, 960], [1, 66, 4, 672, 864], [49, 82, 1052, 672, 960], [1, 674, 676, 96, 192], [1, 674, 1077, 576, 864], [1, 674, 100, 96, 192], [1, 2, 772, 96, 192], [1, 130, 908, 672, 192], [49, 2, 44, 576, 864], [49, 2, 44, 576, 480]], 'aut_group': '4608.lj', 'aut_hash': 3301011005301566777, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4608, 'aut_permdeg': 36, 'aut_perms': [318842221327561257690909875941816386406616, 31303450743034687275934959194218215276611, 248306103963959530579788426022384357112823, 7572557401267206144258658830421212443304, 93306105148693759217543321218895251109913, 291756686910626432258442951697967608255164, 172954022556206567660010165744627783, 3343623553825448687400381976689175316522, 39518024623988977774192303160082597024604, 345308436179273422841365124293065851917675, 345308436181462957899650455838677943488075], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 6, 1, 1], [2, 6, 2, 1], [2, 12, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 3, 2, 1], [4, 6, 1, 1], [4, 6, 2, 3], [4, 12, 1, 3], [6, 2, 1, 1], [6, 2, 2, 1], [6, 6, 1, 2], [6, 6, 2, 1], [6, 8, 1, 1], [6, 12, 2, 2], [6, 16, 1, 2], [6, 16, 2, 1], [8, 6, 4, 1], [8, 18, 4, 1], [8, 36, 4, 2], [12, 2, 2, 2], [12, 6, 2, 2], [12, 8, 2, 1], [12, 12, 2, 6], [12, 16, 1, 1], [12, 16, 2, 2], [24, 48, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5.D_6^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '288.1028', 'autcentquo_hash': 1028, 'autcentquo_nilpotent': False, 'autcentquo_order': 288, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_6\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 3, 2], [2, 6, 3], [2, 12, 1], [3, 2, 1], [3, 8, 1], [3, 16, 1], [4, 1, 2], [4, 2, 1], [4, 3, 2], [4, 6, 7], [4, 12, 3], [6, 2, 3], [6, 6, 4], [6, 8, 1], [6, 12, 4], [6, 16, 4], [8, 6, 4], [8, 18, 4], [8, 36, 8], [12, 2, 4], [12, 6, 4], [12, 8, 2], [12, 12, 12], [12, 16, 5], [24, 48, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '288.1028', 'commutator_count': 1, 'commutator_label': '72.47', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 155870, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['24.12', 1], ['48.10', 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, 3], [2, 12, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 3, 2, 1], [4, 6, 1, 3], [4, 6, 2, 2], [4, 12, 1, 3], [6, 2, 1, 1], [6, 2, 2, 1], [6, 6, 1, 2], [6, 6, 2, 1], [6, 8, 1, 1], [6, 12, 1, 2], [6, 12, 2, 1], [6, 16, 1, 2], [6, 16, 2, 1], [8, 6, 2, 2], [8, 18, 2, 2], [8, 36, 2, 4], [12, 2, 2, 2], [12, 6, 2, 2], [12, 8, 2, 1], [12, 12, 1, 2], [12, 12, 2, 5], [12, 16, 1, 1], [12, 16, 2, 2], [24, 48, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 80640, 'exponent': 24, 'exponents_of_order': [7, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 8]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '288.1028', 'hash': 155870, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 6, 2, 6], 'inner_gens': [[1, 2, 52, 96, 192], [1, 2, 68, 576, 864], [49, 34, 4, 576, 1056], [1, 674, 676, 96, 192], [1, 674, 484, 96, 192]], 'inner_hash': 1028, 'inner_nilpotent': False, 'inner_order': 288, 'inner_split': True, 'inner_tex': 'D_6\\times S_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 6, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 16], [2, 28], [3, 16], [4, 10], [6, 20]], 'label': '1152.155870', 'linC_count': 128, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 128, 'linQ_dim': 9, 'linQ_dim_count': 64, 'linR_count': 32, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3:OD16*S4', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 51, 'number_characteristic_subgroups': 63, 'number_conjugacy_classes': 90, 'number_divisions': 60, 'number_normal_subgroups': 79, 'number_subgroup_autclasses': 502, 'number_subgroup_classes': 614, 'number_subgroups': 3086, 'old_label': None, 'order': 1152, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 39], [3, 26], [4, 88], [6, 150], [8, 384], [12, 272], [24, 192]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 2, 4, 96, 960], [1, 50, 4, 96, 960], [1, 2, 44, 576, 864], [1, 2, 21, 576, 864]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [3, 8], [4, 8], [6, 12], [8, 3], [12, 2], [16, 1], [24, 2]], 'representations': {'PC': {'code': '1325372148165850938277959705989126656129273064907680438904852537122075523081', 'gens': [1, 2, 3, 7, 8], 'pres': [9, -2, -2, -2, -2, -2, -3, -2, 2, -3, 1406, 929, 74, 732, 102, 1813, 130, 1742, 18159, 9096, 5325, 2310, 1374, 31120, 19033, 7810, 1339, 1996, 214, 15578]}, 'Perm': {'d': 15, 'gens': [100677830520, 193158120965, 19200746880, 288450408960, 193158120960, 3, 840, 7, 16]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3:\\OD_{16}\\times S_4', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
