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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '448.564', 'ambient_counter': 564, 'ambient_order': 448, 'ambient_tex': 'C_4^2.D_{14}', 'central': True, 'central_factor': False, 'centralizer_order': 448, 'characteristic': True, 'core_order': 2, 'counter': 105, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '448.564.224.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '224.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '224.137', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 137, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 224, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{28}.D_4', 'simple': True, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '448.564', 'aut_centralizer_order': 10752, 'aut_label': '224.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 10752, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['32.a1.a1', '112.a1.a1', '112.d1.a1', '112.e1.a1'], 'contains': ['448.a1.a1'], 'core': '224.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [659, 770, 9212, 1488, 7639, 3034, 2659, 2387], 'generators': [4], 'label': '448.564.224.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '224.a1.a1', 'normal_contained_in': ['32.a1.a1', '112.a1.a1'], 'normal_contains': ['448.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '224.a1.a1', 'projective_image': '224.137', 'quotient_action_image': '1.1', 'quotient_action_kernel': '224.137', 'quotient_action_kernel_order': 224, 'quotient_fusion': None, 'short_label': '224.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', '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]], 'center_label': '2.1', 'center_order': 2, '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'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 12, 6, 6, 14, 4, 6, 12], 'aut_gens': [[1, 2, 8, 64], [33, 34, 136, 320], [33, 6, 360, 256], [53, 6, 104, 320], [1, 38, 216, 320], [37, 6, 408, 128], [5, 6, 264, 64], [53, 2, 140, 384], [33, 38, 344, 256], [21, 38, 328, 256]], 'aut_group': None, 'aut_hash': 5229004673012084454, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10752, 'aut_permdeg': 68, 'aut_perms': [150305688004633656477897539621775653329665896636594322413811058038058752960911665875088022023031, 788762425728211961755561327935732546708363142863286443013977951447815864832315489442357790330744, 892088883654790749260298803103836821976456265364894584109835233371648027731948827768760044272994, 467407749137593168699923419171324639462416260128274061751212637591753793877994136034193077492130, 836523353544248749837368095373163423388583404843730422768066018635853604144547619196399530911736, 2162854871071466844930909158714892280896772979403729515190231620139166424993327418929012530970947, 2127839777681146882220002320802532794901297701136350228297406739811070246060298209411805349638711, 1199592264324972310691995230763876636725449646918339652912729337290899525093030160301264040291620, 2419803436157842372157617407453823319845659637174093586307394269891511135838946292437976083205259], 'aut_phi_ratio': 56.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 2, 2, 2], [4, 4, 2, 2], [4, 28, 2, 2], [7, 2, 3, 1], [8, 28, 2, 2], [14, 2, 3, 3], [28, 2, 12, 1], [28, 4, 3, 2], [28, 4, 6, 1], [28, 4, 12, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_{14}.(C_2^5\\times C_6).C_2^2', '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], [4, 2, 6], [4, 4, 4], [4, 28, 4], [7, 2, 3], [8, 28, 4], [14, 2, 9], [28, 2, 12], [28, 4, 36]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '112.36', 'commutator_count': 1, 'commutator_label': '28.2', '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': 564, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 2, 2, 2], [4, 4, 1, 2], [4, 4, 2, 1], [4, 28, 1, 2], [4, 28, 2, 1], [7, 2, 3, 1], [8, 28, 1, 2], [8, 28, 2, 1], [14, 2, 3, 3], [28, 2, 12, 1], [28, 4, 3, 2], [28, 4, 6, 3], [28, 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': 564, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 2, 4, 7], 'inner_gens': [[1, 2, 28, 64], [1, 2, 40, 64], [53, 34, 8, 384], [1, 2, 136, 64]], 'inner_hash': 36, 'inner_nilpotent': False, 'inner_order': 112, 'inner_split': True, 'inner_tex': 'C_{14}:D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 52], [4, 14]], 'label': '448.564', 'linC_count': 96, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 32, 'linQ_dim': 16, 'linQ_dim_count': 32, 'linR_count': 36, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.D14', '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': 24, 'number_characteristic_subgroups': 39, 'number_conjugacy_classes': 82, 'number_divisions': 28, 'number_normal_subgroups': 51, 'number_subgroup_autclasses': 92, 'number_subgroup_classes': 108, 'number_subgroups': 388, 'old_label': None, 'order': 448, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 140], [7, 6], [8, 112], [14, 18], [28, 168]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 6], 'outer_gen_pows': [0, 0, 16, 0, 0], 'outer_gens': [[1, 34, 8, 64], [5, 34, 8, 64], [21, 34, 8, 64], [1, 6, 8, 64], [1, 2, 12, 128]], 'outer_group': '96.231', 'outer_hash': 231, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 13, 'outer_perms': [41064, 744, 24, 479001600, 3628803], '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, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 3], [6, 4], [12, 4], [24, 3]], 'representations': {'PC': {'code': 682563215020978787786859352413317, 'gens': [1, 2, 4, 7], 'pres': [7, -2, -2, -2, 2, -2, -2, -7, 224, 36, 787, 570, 80, 1684, 102, 2379]}, 'Perm': {'d': 27, 'gens': [15537113273014441136118213, 2987311597364, 256450679798781448, 122000787836928000, 4410546813871, 5802181616106, 434365636337687549214720000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.D_{14}', 'transitive_degree': 448, '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, 2, 12, 28], 'aut_gens': [[1, 2, 8], [113, 2, 120], [169, 170, 216], [1, 6, 8], [5, 2, 200], [169, 218, 8]], 'aut_group': '2688.dt', 'aut_hash': 8551432399099159210, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2688, 'aut_permdeg': 15, 'aut_perms': [7983360, 87660922138, 479001600, 187293295155, 8025699], 'aut_phi_ratio': 28.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 28, 1, 1], [4, 2, 1, 2], [4, 4, 2, 1], [4, 28, 1, 1], [7, 2, 3, 1], [8, 28, 2, 1], [14, 2, 3, 1], [14, 2, 6, 1], [28, 4, 3, 2], [28, 4, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_7\\times D_4^2', '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': '336.216', 'autcentquo_hash': 216, 'autcentquo_nilpotent': False, 'autcentquo_order': 336, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 28, 1], [4, 2, 2], [4, 4, 2], [4, 28, 1], [7, 2, 3], [8, 28, 2], [14, 2, 9], [28, 4, 18]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '112.36', 'commutator_count': 1, 'commutator_label': '28.2', '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': 137, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 28, 1, 1], [4, 2, 1, 2], [4, 4, 1, 2], [4, 28, 1, 1], [7, 2, 3, 1], [8, 28, 1, 2], [14, 2, 3, 1], [14, 2, 6, 1], [28, 4, 3, 2], [28, 4, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 56, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 0, 6]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '56.12', 'hash': 137, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 4, 14], 'inner_gens': [[1, 6, 120], [117, 2, 216], [113, 18, 8]], 'inner_hash': 36, 'inner_nilpotent': False, 'inner_order': 112, 'inner_split': True, 'inner_tex': 'C_{14}:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 24, 'irrQ_dim': 48, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 26], [4, 7]], 'label': '224.137', 'linC_count': 6, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 4, 'linQ_dim': 14, 'linQ_dim_count': 4, 'linR_count': 3, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C28.D4', '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': 15, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 41, 'number_divisions': 18, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 50, 'number_subgroup_classes': 60, 'number_subgroups': 222, 'old_label': None, 'order': 224, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 31], [4, 40], [7, 6], [8, 56], [14, 18], [28, 72]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 4, 56], 'outer_gens': [[1, 2, 120], [5, 2, 120], [57, 58, 72]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [744, 24, 40323], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\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], [2, 2], [4, 1], [6, 4], [12, 2], [24, 1]], 'representations': {'PC': {'code': 78216869837291616702289465541102021, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -2, -2, -2, -7, 672, 73, 31, 2090, 518, 2883, 2601, 69, 3130, 88, 3467]}, 'Perm': {'d': 23, 'gens': [1134643256731346786047, 2514161715068303596800, 3696112211801025542400, 4752207122534610681600, 5988479609728376046720, 973]}}, '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_{28}.D_4', 'transitive_degree': 112, 'wreath_data': None, 'wreath_product': False}
