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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '128.994', 'ambient_counter': 994, 'ambient_order': 128, 'ambient_tex': 'D_{32}:C_2', 'central': False, 'central_factor': True, 'centralizer_order': 4, 'characteristic': True, 'core_order': 64, 'counter': 8, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '128.994.2.e1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.e1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '64.54', 'subgroup_hash': 54, 'subgroup_order': 64, 'subgroup_tex': 'Q_{64}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '128.994', 'aut_centralizer_order': 2, 'aut_label': '2.e1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '512.58302', 'aut_weyl_index': 2, 'centralizer': '32.b1.a1', 'complements': ['64.b1.a1', '64.c1.b1', '64.c1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.c1.a1', '4.c1.b1', '4.d1.a1'], 'core': '2.e1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2807, 5499, 3484, 4012, 2733, 5616, 3484, 3973], 'generators': [58411119, 47459024, 20078881, 87616704, 7301469, 903361], 'label': '128.994.2.e1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.e1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.c1.a1', '4.c1.b1', '4.d1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.e1.a1', 'projective_image': '64.186', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.e1.a1', 'subgroup_fusion': None, 'weyl_group': '32.18'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 32, 'aut_gen_orders': [2, 8, 32], 'aut_gens': [[1, 2], [31, 62], [21, 6], [51, 34]], 'aut_group': '512.58302', 'aut_hash': 7772715459544409331, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 32, 'aut_perms': [209110625063666949217196394943844509, 166831302240608821283181734550591682, 192584446486825221723391051205525821], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 16, 2, 1], [8, 2, 2, 1], [16, 2, 4, 1], [32, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{32}:C_8', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 16, 'autcentquo_group': '128.913', 'autcentquo_hash': 913, 'autcentquo_nilpotent': True, 'autcentquo_order': 128, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{16}:C_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 16, 2], [8, 2, 2], [16, 2, 4], [32, 2, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '32.18', 'commutator_count': 1, 'commutator_label': '16.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 54, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 16, 1, 2], [8, 2, 2, 1], [16, 2, 4, 1], [32, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 32, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, -1, 8]], 'familial': True, 'frattini_label': '16.1', 'frattini_quotient': '4.2', 'hash': 54, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 16, 'inner_gen_orders': [2, 16], 'inner_gens': [[1, 62], [5, 2]], 'inner_hash': 18, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'D_{16}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 32, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 15]], 'label': '64.54', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 32, 'linQ_dim_count': 1, 'linR_count': 8, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'Q64', 'ngens': 2, 'nilpotency_class': 5, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 19, 'number_divisions': 8, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 11, 'number_subgroup_classes': 15, 'number_subgroups': 37, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 34], [8, 4], [16, 8], [32, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 8, 'outer_gen_orders': [2, 8], 'outer_gen_pows': [2, 0], 'outer_gens': [[3, 2], [1, 10]], 'outer_group': '16.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 10, 'outer_perms': [362880, 6057], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_8', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 64, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1], [8, 1], [16, 1]], 'representations': {'PC': {'code': 309504273051162771967, 'gens': [1, 2], 'pres': [6, -2, 2, -2, -2, -2, -2, 192, 745, 31, 1082, 50, 1347, 69, 1444, 88]}, 'Lie': [{'d': 2, 'q': 31, 'gens': [896293, 653851, 893760, 405374], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 31, 'gens': [893666, 687030]}, 'Perm': {'d': 64, 'gens': [66450134624445933678799560050331607746391532584685435254529204790382042061733888000000000, 34216625329393954566794504615428400130324458319121250657957939996263067413000027270329600, 24175131827739636274127375045840733860693888662420563914564207161518320707982436785337600, 12087040917006882684584735480871444149038188995887599735100184001816510068360415118652856, 6010765313484612917937180565556780809330360821355660294198753718800751569568892186006982, 1983116034341975454789449189498620955194254460449807567074683265838922666583379392681647]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_{64}', 'transitive_degree': 64, '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': 32, 'aut_gen_orders': [2, 2, 8, 32], 'aut_gens': [[47459024, 9506, 87616609], [47459024, 9506, 912769], [62974482, 750101, 87616609], [50197045, 315347, 912769], [41070354, 191478, 87616609]], 'aut_group': '1024.gd', 'aut_hash': 3877486645027153190, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 1024, 'aut_permdeg': 34, 'aut_perms': [1, 94652509007419664227072453082613496728, 97804868215586374576120373477885971464, 58831109592893853197971941825263618815], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 16, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 16, 2, 1], [8, 2, 2, 2], [16, 2, 4, 2], [32, 2, 8, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_{32}:C_8', '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': 16, 'autcentquo_group': '128.913', 'autcentquo_hash': 913, 'autcentquo_nilpotent': True, 'autcentquo_order': 128, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{16}:C_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 16, 2], [4, 1, 2], [4, 2, 1], [4, 16, 2], [8, 2, 4], [16, 2, 8], [32, 2, 16]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '32.18', 'commutator_count': 1, 'commutator_label': '16.1', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 994, '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, 16, 1, 2], [4, 1, 2, 1], [4, 2, 1, 1], [4, 16, 1, 2], [8, 2, 2, 2], [16, 2, 4, 2], [32, 2, 8, 2]], 'element_repr_type': 'GLFp', 'elementary': 2, 'eulerian_function': 672, 'exponent': 32, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 16]], 'familial': False, 'frattini_label': '16.1', 'frattini_quotient': '8.5', 'hash': 994, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 16, 'inner_gen_orders': [16, 2, 2], 'inner_gens': [[47459024, 83517, 87616609], [25554896, 9506, 912769], [47459024, 912576, 87616609]], 'inner_hash': 18, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'D_{16}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 30]], 'label': '128.994', 'linC_count': 16, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 32, 'linQ_degree_count': 1, 'linQ_dim': 32, 'linQ_dim_count': 1, 'linR_count': 8, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D32:C2', 'ngens': 3, 'nilpotency_class': 5, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 38, 'number_divisions': 15, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 53, 'number_subgroups': 172, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 35], [4, 36], [8, 8], [16, 16], [32, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 8, 'outer_gen_orders': [2, 2, 8], 'outer_gen_pows': [912674, 25554896, 912674], 'outer_gens': [[47459024, 9506, 912769], [47459024, 491984, 87616609], [27380245, 9506, 87616609]], 'outer_group': '32.36', 'outer_hash': 36, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 12, 'outer_perms': [362880, 39916800, 6057], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_8', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 64, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 2], [8, 2], [32, 1]], 'representations': {'PC': {'code': 19497536261188101861280734, 'gens': [1, 2, 7], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 869, 36, 1262, 58, 1571, 80, 1684, 102, 4710]}, 'GLFp': {'d': 2, 'p': 97, 'gens': [47459024, 9506, 87616609]}, 'Perm': {'d': 64, 'gens': [64467018590103958224010110860832986791197278124235627695686212845121345606196920320000000, 32233509295051979112005055426188522099310973521152144973615375142089828857365032918201600, 4028664439651445760304855308136396429713341927997617401657973553214258184617830699100937, 24175131827739636274127375045840733860693888662420563914564207161518320707982436785337600, 12087040917006882684584735480871444149038188995887599735100184001816510068360415118652856, 6010765313484612917937180565556780809330360821355660294198753718800751569568892186006982, 1983116034341975454789449189498620955194254460449807567074683265838922666583379392681647]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{32}:C_2', 'transitive_degree': 64, 'wreath_data': None, 'wreath_product': False}
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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}
