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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1568.405', 'ambient_counter': 405, 'ambient_order': 1568, 'ambient_tex': 'C_{56}.D_{14}', 'central': False, 'central_factor': False, 'centralizer_order': 14, 'characteristic': False, 'core_order': 98, 'counter': 27, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1568.405.8.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '8.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': 8, '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': '196.5', 'subgroup_hash': 5, 'subgroup_order': 196, 'subgroup_tex': 'C_7:C_{28}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1568.405', 'aut_centralizer_order': 56, 'aut_label': '8.f1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '504.178', 'aut_weyl_index': 112, 'centralizer': '112.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.d1.a1', '4.h1.a1', '4.j1.a1'], 'contains': ['16.a1.a1', '56.m1.a1', '56.n1.a1'], 'core': '16.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [2180, -1, 2102, -1, 8703, -1, 1155, -1], 'generators': [589, 784, 4, 224], 'label': '1568.405.8.f1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '8.f1.a1', 'projective_image': '784.125', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.f1.a1', 'subgroup_fusion': None, 'weyl_group': '56.12'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '28.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [2, 6, 42], 'aut_gens': [[1, 28], [15, 28], [1, 84], [73, 28]], 'aut_group': '504.178', 'aut_hash': 178, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 504, 'aut_permdeg': 14, 'aut_perms': [744, 526943764, 19806758163], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 7, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 18, 1], [14, 1, 6, 1], [14, 2, 3, 1], [14, 2, 18, 1], [28, 7, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{14}:C_6^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_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, 1], [4, 7, 2], [7, 1, 6], [7, 2, 21], [14, 1, 6], [14, 2, 21], [28, 7, 12]], 'center_label': '14.2', 'center_order': 14, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '7.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['28.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 7, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 6, 3], [14, 1, 6, 1], [14, 2, 3, 1], [14, 2, 6, 3], [28, 7, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 28, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[2, 0, 18]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '98.3', 'hash': 5, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 168], [57, 28]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 28], [2, 42]], 'label': '196.5', 'linC_count': 18, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 5, 'linQ_dim': 12, 'linQ_dim_count': 3, 'linR_count': 27, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C7:C28', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 70, 'number_divisions': 14, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 18, 'number_subgroups': 36, 'old_label': None, 'order': 196, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 14], [7, 48], [14, 48], [28, 84]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[11, 28], [13, 84]], 'outer_group': '36.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 36, 'outer_permdeg': 10, 'outer_perms': [362883, 5184], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [4, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 4], [12, 7]], 'representations': {'PC': {'code': 128398020947, 'gens': [1, 4], 'pres': [4, -2, -2, -7, -7, 8, 21, 2691]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [390225, 536562, 957]}, 'Perm': {'d': 18, 'gens': [4032127, 378011776665600, 7983360, 973]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [28], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7:C_{28}', 'transitive_degree': 28, '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}
