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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '448.720', 'ambient_counter': 720, 'ambient_order': 448, 'ambient_tex': 'D_{14}:Q_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 224, 'characteristic': True, 'core_order': 4, 'counter': 130, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '448.720.112.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '112.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '112.31', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 31, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 112, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4\\times D_7', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '448.720', 'aut_centralizer_order': 5376, 'aut_label': '112.b1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 5376, 'centralizer': '2.c1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.c1.a1', '56.a1.a1', '56.i1.a1', '56.l1.a1', '56.m1.a1', '56.n1.a1', '56.n1.b1', '56.o1.a1'], 'contains': ['224.c1.a1'], 'core': '112.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4084, 4307, 4763, 4798, 7106, 5139, 6434, 4605], 'generators': [114], 'label': '448.720.112.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '112.b1.a1', 'normal_contained_in': ['16.c1.a1', '56.a1.a1'], 'normal_contains': ['224.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '112.b1.a1', 'projective_image': '224.132', 'quotient_action_image': '2.1', 'quotient_action_kernel': '56.5', 'quotient_action_kernel_order': 56, 'quotient_fusion': None, 'short_label': '112.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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, 1], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, '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': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, '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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 28, 12, 2, 12, 6, 28, 6], 'aut_gens': [[1, 2, 4, 8], [227, 2, 230, 408], [225, 2, 230, 362], [291, 2, 340, 232], [161, 2, 342, 296], [289, 2, 4, 216], [257, 2, 340, 298], [129, 2, 230, 42], [419, 2, 342, 232], [321, 2, 6, 72]], 'aut_group': None, 'aut_hash': 5229004673012084454, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10752, 'aut_permdeg': 44, 'aut_perms': [863642516126698202437149292294677568028215919124439676, 11938250200722953547271067078465762183241087467888234, 2389805197059983963760613800991254607771188181590968112, 2630929951295667896864568407899905540601729202216746664, 2373354926503922237639252956674293281179322825973654062, 2610798587931555351333482965321150368811327267379966465, 11636728703221028378186416638708834273939192994090856, 2631097275122875839817801640133697702715882695020272984, 364199559471178095398973923848835728076399567639489739], 'aut_phi_ratio': 56.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 14, 2, 1], [4, 2, 1, 2], [4, 4, 2, 1], [4, 8, 1, 1], [4, 28, 1, 1], [4, 28, 2, 1], [4, 56, 1, 1], [7, 2, 3, 1], [8, 4, 2, 1], [8, 28, 2, 1], [14, 2, 3, 3], [28, 4, 3, 2], [28, 8, 6, 2], [56, 4, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_7.(C_2^4\\times C_6).C_2^4', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 14, 2], [4, 2, 2], [4, 4, 2], [4, 8, 1], [4, 28, 3], [4, 56, 1], [7, 2, 3], [8, 4, 2], [8, 28, 2], [14, 2, 9], [28, 4, 6], [28, 8, 12], [56, 4, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '112.31', 'commutator_count': 1, 'commutator_label': '56.8', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 720, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 14, 1, 2], [4, 2, 1, 2], [4, 4, 1, 2], [4, 8, 1, 1], [4, 28, 1, 3], [4, 56, 1, 1], [7, 2, 3, 1], [8, 4, 2, 1], [8, 28, 2, 1], [14, 2, 3, 3], [28, 4, 3, 2], [28, 8, 3, 2], [28, 8, 6, 1], [56, 4, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2688, 'exponent': 56, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '56.12', 'hash': 720, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 1, 2, 28], 'inner_gens': [[1, 2, 6, 330], [1, 2, 4, 8], [3, 2, 4, 120], [131, 2, 340, 8]], 'inner_hash': 31, 'inner_nilpotent': False, 'inner_order': 112, 'inner_split': None, 'inner_tex': 'D_4\\times D_7', 'inner_used': [1, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 34], [4, 19]], 'label': '448.720', 'linC_count': 48, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 48, 'linQ_dim': 16, 'linQ_dim_count': 48, 'linR_count': 228, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D14:Q16', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 23, 'number_characteristic_subgroups': 37, 'number_conjugacy_classes': 61, 'number_divisions': 27, 'number_normal_subgroups': 45, 'number_subgroup_autclasses': 116, 'number_subgroup_classes': 148, 'number_subgroups': 740, 'old_label': None, 'order': 448, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 31], [4, 160], [7, 6], [8, 64], [14, 18], [28, 120], [56, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 6], 'outer_gen_pows': [80, 400, 272, 0, 336], 'outer_gens': [[419, 2, 342, 232], [291, 2, 340, 232], [161, 2, 342, 232], [289, 2, 4, 216], [257, 2, 340, 298]], 'outer_group': '96.231', 'outer_hash': 231, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 96, 'outer_perms': [84292983624727090575631421500974168923399281113415589031874962534759771259907700071527350360883175659423665215786655809077972738267859557156851906535, 31869085030745359201185193160801615619941701608251184887270194691324229390312633249570448079540266523868930636539707772172535540485534656115357269760, 406103710694036172467297631855986329156754648233509971783270350327587084447316809242961008151913119021271317397970404126171028629028456074622651673534, 52973491932398537246409076010213363829232779895138756153755514765197651735586361338476155034796329164322386420595254354681607094692171764603220288000, 63413376420789604158375763309876763762846183420699860860636401147674885383682733268689649296334002824022012611479483054260655075910944463668668260401], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 3], [6, 4], [12, 4], [24, 2]], 'representations': {'PC': {'code': 122963815830230947244470511914059082074629, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, 2, 2, -2, -2, -7, 128, 1192, 9243, 857, 80, 7284, 2118, 102, 17477, 124, 18822]}, 'Perm': {'d': 27, 'gens': [15537113273012830507008000, 256096558858232133, 13163422191067484, 256094948229120000, 4404319792351, 5808408640626, 434365636337687549214720000]}}, 'schur_multiplier': [2, 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_{14}:Q_{16}', 'transitive_degree': 224, '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': 84, 'aut_gen_orders': [2, 2, 6, 28], 'aut_gens': [[1, 2, 4], [57, 58, 60], [57, 2, 4], [1, 2, 68], [97, 30, 4]], 'aut_group': '672.1093', 'aut_hash': 1093, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 672, 'aut_permdeg': 13, 'aut_perms': [120, 1, 44357040, 2402270934], 'aut_phi_ratio': 14.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [2, 7, 2, 1], [2, 14, 2, 1], [4, 2, 1, 1], [4, 14, 1, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 4, 6, 1], [28, 4, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4\\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': '84.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 84, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [2, 7, 2], [2, 14, 2], [4, 2, 1], [4, 14, 1], [7, 2, 3], [14, 2, 3], [14, 4, 6], [28, 4, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '56.12', 'commutator_count': 1, 'commutator_label': '14.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [2, 7, 1, 2], [2, 14, 1, 2], [4, 2, 1, 1], [4, 14, 1, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 4, 3, 2], [28, 4, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 672, 'exponent': 28, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 1, 3]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '56.12', 'hash': 31, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 2, 14], 'inner_gens': [[1, 2, 52], [1, 2, 60], [65, 58, 4]], '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': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 14], [4, 3]], 'label': '112.31', 'linC_count': 27, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 8, 'linR_count': 27, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4*D7', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 25, 'number_divisions': 15, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 34, 'number_subgroup_classes': 54, 'number_subgroups': 220, 'old_label': None, 'order': 112, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 47], [4, 16], [7, 6], [14, 30], [28, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [28, 0], 'outer_gens': [[1, 30, 4], [57, 2, 20]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [6, 4], [12, 1]], 'representations': {'PC': {'code': 22553896189964738693, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -7, 782, 457, 42, 2083, 58, 2404]}, 'GLZN': {'d': 2, 'p': 21, 'gens': [142311, 9325, 9269, 74096, 59004]}, 'Perm': {'d': 11, 'gens': [368047, 1, 16, 7, 4364760]}}, 'schur_multiplier': [2, 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_4\\times D_7', 'transitive_degree': 28, 'wreath_data': None, 'wreath_product': False}
