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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '959.1', 'ambient_counter': 1, 'ambient_order': 959, 'ambient_tex': 'C_{959}', 'central': True, 'central_factor': False, 'centralizer_order': 959, 'characteristic': True, 'core_order': 137, 'counter': 2, 'cyclic': True, 'direct': True, 'hall': 137, 'label': '959.1.7.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '7.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '7.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 7, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_7', 'simple': True, 'solvable': True, 'special_labels': ['C1'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '137.1', 'subgroup_hash': 1, 'subgroup_order': 137, 'subgroup_tex': 'C_{137}', 'supersolvable': True, 'sylow': 137}
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gps_subgroup_data • Show schema
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{'ambient': '959.1', 'aut_centralizer_order': 6, 'aut_label': '7.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '136.2', 'aut_weyl_index': 6, 'centralizer': '1.a1.a1', 'complements': ['137.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['959.a1.a1'], 'core': '7.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7711, 7711, 2742, 2742, 7393, 7393, 2607, 2607], 'generators': [7], 'label': '959.1.7.a1.a1', 'mobius_quo': -1, 'mobius_sub': -1, 'normal_closure': '7.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['959.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '7.a1.a1', 'projective_image': '7.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '7.1', 'quotient_action_kernel_order': 7, 'quotient_fusion': None, 'short_label': '7.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': '137.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 136, 'aut_gen_orders': [136], 'aut_gens': [[1], [79]], 'aut_group': '136.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 136, 'aut_permdeg': 25, 'aut_perms': [647478071426320662528000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [137, 1, 136, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{136}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 136, 'autcent_group': '136.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 136, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{136}', '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], [137, 1, 136]], 'center_label': '137.1', 'center_order': 137, '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': ['137.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [137, 1, 136, 1]], 'element_repr_type': 'PC', 'elementary': 137, 'eulerian_function': 1, 'exponent': 137, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 17], 'factors_of_order': [137], 'faithful_reps': [[1, 0, 136]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '137.1', 'hash': 1, 'hyperelementary': 137, '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': 136, 'irrQ_dim': 136, 'irrR_degree': 2, 'irrep_stats': [[1, 137]], 'label': '137.1', 'linC_count': 136, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 136, 'linQ_degree_count': 1, 'linQ_dim': 136, 'linQ_dim_count': 1, 'linR_count': 68, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C137', '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': 137, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 137, 'order_factorization_type': 1, 'order_stats': [[1, 1], [137, 136]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 136, 'outer_gen_orders': [136], 'outer_gen_pows': [0], 'outer_gens': [[79]], 'outer_group': '136.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 136, 'outer_permdeg': 25, 'outer_perms': [647478071426320662528000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{136}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 137, 'pgroup': 137, 'primary_abelian_invariants': [137], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [136, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -137]}, 'Lie': [{'d': 1, 'q': 137, 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 137, 'gens': [2571491]}, 'Perm': {'d': 137, 'gens': [4976298319455466174458029019907054949782919521446938757232838853791105783164137017260699982630670355464999711336810105210045801809228929514064353192677551019303997993620395950915549548359646418539905024000000000000000000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [137], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{137}', 'transitive_degree': 137, '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': '959.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 408, 'aut_gen_orders': [2, 408], 'aut_gens': [[1], [412], [627]], 'aut_group': '816.84', 'aut_hash': 84, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 816, 'aut_permdeg': 142, 'aut_perms': [5735105786351573714773333379777534119621557357369519893687393866638222204817005861172274278713100812751121968249096854749672390264183999584665085346022690145839471639273436334239723954097376284516582757821841408000000000000000000000000000000000, 7646423086216498797700107684393860082011467955946672780535082266667398769110708643985740883289493881625490270744134079943956785673071769690104048545043279815979862486538495620568292385829449228507521935421211432739437325089958173396172800000000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [7, 1, 6, 1], [137, 1, 136, 1], [959, 1, 816, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{408}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 408, 'autcent_group': '816.84', 'autcent_hash': 84, 'autcent_nilpotent': True, 'autcent_order': 816, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{408}', '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], [7, 1, 6], [137, 1, 136], [959, 1, 816]], 'center_label': '959.1', 'center_order': 959, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['7.1', '137.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['137.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [7, 1, 6, 1], [137, 1, 136, 1], [959, 1, 816, 1]], 'element_repr_type': 'PC', 'elementary': 959, 'eulerian_function': 1, 'exponent': 959, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3, 17], 'factors_of_order': [7, 137], 'faithful_reps': [[1, 0, 816]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '959.1', 'hash': 1, 'hyperelementary': 959, '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': 816, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 959]], 'label': '959.1', 'linC_count': None, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C959', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 959, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 959, 'order_factorization_type': 11, 'order_stats': [[1, 1], [7, 6], [137, 136], [959, 816]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 408, 'outer_gen_orders': [2, 408], 'outer_gen_pows': [0, 0], 'outer_gens': [[412], [627]], 'outer_group': '816.84', 'outer_hash': 84, 'outer_nilpotent': True, 'outer_order': 816, 'outer_permdeg': 142, 'outer_perms': [5735105786351573714773333379777534119621557357369519893687393866638222204817005861172274278713100812751121968249096854749672390264183999584665085346022690145839471639273436334239723954097376284516582757821841408000000000000000000000000000000000, 7646423086216498797700107684393860082011467955946672780535082266667398769110708643985740883289493881625490270744134079943956785673071769690104048545043279815979862486538495620568292385829449228507521935421211432739437325089958173396172800000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{408}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 144, 'pgroup': 0, 'primary_abelian_invariants': [7, 137], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 1], [6, 1], [136, 1], [816, 1]], 'representations': {'PC': {'code': 93295, 'gens': [1], 'pres': [2, -7, -137, 14]}, 'GLFp': {'d': 2, 'p': 547, 'gens': [53470865070]}, 'Perm': {'d': 144, 'gens': [231262243030804366231293944189599504916665930749281786305961287657217072521162793508865080999953922191792901790315780178107215006221742430911621152128895666016996929858043911325249033271860782931562015989308420259840000000000000000000000000000000000, 4976298319455466174458029019907054949782919521446938757232838853791105783164137017260699982630670355464999711336810105210045801809228929514064353192677551019303997993620395950915549548359646418539905024000000000000000000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [959], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{959}', 'transitive_degree': 959, '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': '7.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [6], 'aut_gens': [[1], [3]], 'aut_group': '6.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 6, 'aut_permdeg': 5, 'aut_perms': [27], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [7, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', '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], [7, 1, 6]], 'center_label': '7.1', 'center_order': 7, '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': ['7.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [7, 1, 6, 1]], 'element_repr_type': 'PC', 'elementary': 7, 'eulerian_function': 1, 'exponent': 7, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [7], 'faithful_reps': [[1, 0, 6]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '7.1', 'hash': 1, 'hyperelementary': 7, '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': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 7]], 'label': '7.1', 'linC_count': 6, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C7', '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': 7, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 7, 'order_factorization_type': 1, 'order_stats': [[1, 1], [7, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 7, 'primary_abelian_invariants': [7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [6, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -7]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [74473229056828135]}, 'Lie': [{'d': 1, 'q': 7, 'gens': [873], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 7, 'gens': [351]}, 'Perm': {'d': 7, 'gens': [4320]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [7], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7', 'transitive_degree': 7, 'wreath_data': None, 'wreath_product': False}
