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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1568.405', 'ambient_counter': 405, 'ambient_order': 1568, 'ambient_tex': 'C_{56}.D_{14}', 'central': False, 'central_factor': False, 'centralizer_order': 784, 'characteristic': True, 'core_order': 4, 'counter': 140, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1568.405.392.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '392.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '392.41', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 41, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 392, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_7\\times D_{14}', 'simple': False, 'solvable': True, 'special_labels': ['Phi', 'U1'], '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': '1568.405', 'aut_centralizer_order': 28224, 'aut_label': '392.a1', 'aut_quo_index': 8, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 28224, 'centralizer': '2.c1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['56.b1.a1', '56.c1.a1', '56.i1.a1', '56.i1.b1', '56.i1.c1', '196.a1.a1', '196.c1.a1', '196.f1.a1', '196.g1.a1', '196.h1.a1', '196.i1.a1', '196.j1.a1'], 'contains': ['784.a1.a1'], 'core': '392.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6727, 5309, 7130, 4691, 3902, 3876, 6544, 3920], 'generators': [392], 'label': '1568.405.392.a1.a1', 'mobius_quo': 0, 'mobius_sub': -392, 'normal_closure': '392.a1.a1', 'normal_contained_in': ['56.b1.a1', '56.c1.a1', '196.a1.a1'], 'normal_contains': ['784.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '392.a1.a1', 'projective_image': '784.125', 'quotient_action_image': '2.1', 'quotient_action_kernel': '196.10', 'quotient_action_kernel_order': 196, 'quotient_fusion': None, 'short_label': '392.a1.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, 12, 42, 12, 6, 12, 12], 'aut_gens': [[1, 2, 28], [25, 250, 1260], [409, 454, 1260], [801, 562, 1316], [1201, 1258, 700], [405, 1014, 1428], [401, 1014, 364], [1197, 582, 1484]], 'aut_group': None, 'aut_hash': 2901852258261930156, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 56448, 'aut_permdeg': 46, 'aut_perms': [1814630638113581699078267659529966240108866667847251773594, 4098755172961351082247060920971773774585545747211818307886, 2771555209590465402358553080369999444608764298359432161155, 4597377737350875030363450528301471612116609399445528680896, 4440174530090327356252931262435425383507637283707252087686, 5203229419751955342798767117943373425740028490886780559748, 5416984104281891547150453046457653325450416658966697902136], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 14, 1, 1], [2, 196, 1, 1], [4, 2, 1, 1], [4, 14, 1, 1], [4, 28, 1, 2], [4, 196, 1, 1], [7, 2, 3, 2], [7, 4, 9, 1], [8, 4, 1, 1], [8, 28, 1, 1], [14, 2, 3, 2], [14, 4, 9, 1], [14, 28, 3, 1], [28, 2, 6, 1], [28, 4, 3, 1], [28, 4, 18, 1], [28, 28, 3, 1], [28, 56, 3, 2], [56, 4, 6, 2], [56, 4, 36, 1], [56, 28, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_7^2.C_6^2.C_2^5', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': None, 'autcentquo_hash': 8777691769077315760, 'autcentquo_nilpotent': False, 'autcentquo_order': 7056, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_7^2.C_6^2.C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 14, 1], [2, 196, 1], [4, 2, 1], [4, 14, 1], [4, 28, 2], [4, 196, 1], [7, 2, 6], [7, 4, 9], [8, 4, 1], [8, 28, 1], [14, 2, 6], [14, 4, 9], [14, 28, 3], [28, 2, 6], [28, 4, 21], [28, 28, 3], [28, 56, 6], [56, 4, 48], [56, 28, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '784.125', 'commutator_count': 1, 'commutator_label': '196.7', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '7.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 405, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 14, 1, 1], [2, 196, 1, 1], [4, 2, 1, 1], [4, 14, 1, 1], [4, 28, 1, 2], [4, 196, 1, 1], [7, 2, 3, 2], [7, 4, 3, 3], [8, 4, 1, 1], [8, 28, 1, 1], [14, 2, 3, 2], [14, 4, 3, 3], [14, 28, 3, 1], [28, 2, 6, 1], [28, 4, 3, 1], [28, 4, 6, 3], [28, 28, 3, 1], [28, 56, 3, 2], [56, 4, 6, 2], [56, 4, 12, 3], [56, 28, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 21504, 'exponent': 56, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 0, 36]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '392.41', 'hash': 405, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 14, 28], 'inner_gens': [[1, 26, 420], [5, 2, 364], [1177, 1234, 28]], 'inner_hash': 125, 'inner_nilpotent': False, 'inner_order': 784, 'inner_split': True, 'inner_tex': 'D_7\\times D_{28}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 38], [4, 88]], 'label': '1568.405', 'linC_count': 36, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 22, 'linQ_dim': 20, 'linQ_dim_count': 22, 'linR_count': 18, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C56.D14', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 28, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 134, 'number_divisions': 36, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 130, 'number_subgroup_classes': 150, 'number_subgroups': 2262, 'old_label': None, 'order': 1568, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 211], [4, 268], [7, 48], [8, 32], [14, 132], [28, 516], [56, 360]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6, 6], 'outer_gen_pows': [0, 392, 0], 'outer_gens': [[785, 26, 812], [1177, 18, 1484], [1, 6, 28]], 'outer_group': '72.50', 'outer_hash': 50, 'outer_nilpotent': True, 'outer_order': 72, 'outer_permdeg': 12, 'outer_perms': [5040, 362883, 39916944], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 1], [6, 8], [12, 9], [24, 5], [48, 3]], 'representations': {'PC': {'code': 41526343046935395249028181008387440113393539396713941, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -7, -2, -2, -2, -7, 5488, 365, 36, 506, 11763, 5106, 80, 29404, 12751, 102, 30588, 124, 32941]}, 'Perm': {'d': 30, 'gens': [8886621040591523400733018468675, 8865560047526522674543087560273, 3017977412900, 4416697629436, 5794513930183, 18621643860556954568330563584000, 27801770031432863891039526912000]}}, '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': 'C_{56}.D_{14}', 'transitive_degree': 112, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, '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': 84, 'aut_gen_orders': [6, 4, 6, 14, 7], 'aut_gens': [[1, 2, 28], [210, 85, 208], [266, 289, 220], [209, 258, 140], [1, 54, 28], [21, 2, 28]], 'aut_group': '14112.ba', 'aut_hash': 3417587930551382414, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14112, 'aut_permdeg': 28, 'aut_perms': [215568105561113732252856929709, 253633904446892962488303547996, 79371368704849436434859931105, 533156523385739650171201341, 218198780472572310805325509387], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 4, 1], [2, 49, 2, 1], [7, 2, 6, 1], [7, 4, 9, 1], [14, 2, 6, 1], [14, 4, 9, 1], [14, 14, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_7^2:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '3528.br', 'autcentquo_hash': 519392570028930314, 'autcentquo_nilpotent': False, 'autcentquo_order': 3528, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_7\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 7, 4], [2, 49, 2], [7, 2, 6], [7, 4, 9], [14, 2, 6], [14, 4, 9], [14, 14, 12]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '196.9', 'commutator_count': 1, 'commutator_label': '49.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '7.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 41, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 2], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 1, 4], [2, 49, 1, 2], [7, 2, 3, 2], [7, 4, 3, 3], [14, 2, 3, 2], [14, 4, 3, 3], [14, 14, 3, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 14, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 1, 9]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '392.41', 'hash': 41, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 14, 7], 'inner_gens': [[1, 26, 28], [5, 2, 364], [1, 58, 28]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 196, 'inner_split': True, 'inner_tex': 'D_7^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 24], [4, 18]], 'label': '392.41', 'linC_count': 117, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 15, 'linQ_dim': 12, 'linQ_dim_count': 15, 'linR_count': 117, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D7*D14', 'ngens': 5, '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': 9, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 50, 'number_divisions': 22, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 79, 'number_subgroups': 754, 'old_label': None, 'order': 392, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 127], [7, 48], [14, 216]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [6, 2, 6], 'outer_gen_pows': [0, 15, 0], 'outer_gens': [[197, 206, 84], [210, 141, 204], [197, 26, 84]], 'outer_group': '72.30', 'outer_hash': 30, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 10, 'outer_perms': [458656, 848165, 15863], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\wr C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [6, 8], [12, 6]], 'representations': {'PC': {'code': 5914847319693657449081, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -7, -2, -7, 261, 26, 362, 3648, 58, 4209]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [25677558, 34720862, 34603218, 473244, 4590980]}, 'Perm': {'d': 16, 'gens': [87660921600, 127, 40320, 973, 1482992582400]}}, '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_7\\times D_{14}', 'transitive_degree': 28, 'wreath_data': None, 'wreath_product': False}
