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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '3072.cc', 'ambient_counter': 55, 'ambient_order': 3072, 'ambient_tex': '(C_4\\times C_8).\\GL(2,\\mathbb{Z}/4)', 'central': False, 'central_factor': False, 'centralizer_order': 512, 'characteristic': True, 'core_order': 64, 'counter': 294, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '3072.cc.48.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '48.g1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '48.9', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 9, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 48, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_6:C_8', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '64.246', 'subgroup_hash': 246, 'subgroup_order': 64, 'subgroup_tex': 'C_2^3\\times C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3072.cc', 'aut_centralizer_order': None, 'aut_label': '48.g1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '6.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.d1.a1', '24.c1.a1', '24.e1.a1', '24.e1.b1'], 'contains': ['96.f1.a1', '96.q1.a1', '96.s1.a1', '96.bj1.a1', '96.bl1.a1', '96.eg1.a1', '96.ei1.a1'], 'core': '48.g1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [631579, 49665, 1015839, 573457], 'label': '3072.cc.48.g1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '48.g1.a1', 'normal_contained_in': ['16.d1.a1', '24.c1.a1', '24.e1.b1', '24.e1.a1'], 'normal_contains': ['96.f1.a1', '96.s1.a1', '96.q1.a1', '192.e1.a1'], 'normalizer': '1.a1.a1', 'old_label': '48.g1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '48.g1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '64.246', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 168, 'aut_gen_orders': [12, 6], 'aut_gens': [[1, 2, 4, 8], [38, 5, 1, 60], [6, 3, 4, 43]], 'aut_group': '43008.cc', 'aut_hash': 6154471930669946344, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 43008, 'aut_permdeg': 18, 'aut_perms': [1183796452478569, 372928794663960], 'aut_phi_ratio': 1344.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [4, 1, 2, 1], [4, 1, 14, 1], [8, 1, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2.C_2^7:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': '43008.cc', 'autcent_hash': 6154471930669946344, 'autcent_nilpotent': False, 'autcent_order': 43008, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2.C_2^7:\\GL(3,2)', '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, 15], [4, 1, 16], [8, 1, 32]], 'center_label': '64.246', 'center_order': 64, 'central_product': True, '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', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 246, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [4, 1, 2, 8], [8, 1, 4, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 120, 'exponent': 8, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '16.14', 'hash': 246, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]], '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, 64]], 'label': '64.246', 'linC_count': None, 'linC_degree': 4, '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': False, 'monomial': True, 'name': 'C2^3*C8', 'ngens': 4, 'nilpotency_class': 1, '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 64, 'number_divisions': 32, 'number_normal_subgroups': 169, 'number_subgroup_autclasses': 19, 'number_subgroup_classes': 169, 'number_subgroups': 169, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15], [4, 16], [8, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [12, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[38, 5, 1, 60], [6, 3, 4, 43]], 'outer_group': '43008.cc', 'outer_hash': 6154471930669946344, 'outer_nilpotent': False, 'outer_order': 43008, 'outer_permdeg': 18, 'outer_perms': [1183796452478569, 372928794663960], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2.C_2^7:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 8], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 8]], 'representations': {'PC': {'code': 69206040, 'gens': [1, 2, 3, 4], 'pres': [6, -2, 2, 2, 2, -2, -2, 69, 88]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [12417, 2049, 8645, 9889, 2043, 1801]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [28679, 4169, 36873, 4115, 61475, 4225]}, 'Perm': {'d': 14, 'gens': [40176, 362880, 6227020800, 39916800, 16582, 5167]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 8], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3\\times C_8', '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': '32.3', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 48, 'aut_gen_orders': [8, 12, 12, 4, 4, 12, 16, 12], 'aut_gens': [[591870, 236718, 49665, 1015839, 573457], [212778, 93245, 49665, 1015839, 557585], [1043921, 574578, 573457, 1015839, 557585], [122317, 126014, 573457, 1015839, 557585], [48271, 614073, 573457, 1015839, 49665], [536032, 72873, 49665, 491535, 557585], [249801, 524222, 573457, 1015839, 557585], [9392, 575074, 49665, 1015839, 573457], [834039, 253630, 557585, 491535, 49665]], 'aut_group': None, 'aut_hash': 5314746377555349766, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 98304, 'aut_permdeg': 768, 'aut_perms': 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841303265433437328376376945895559522122940655844490035610447737820804434065698864701375073079495589870948954549655527453096937735752733137620235044404864239011822877913196015709078072866314267306838397211634893870342986371353020573066074970021319661852531093128296326488186083215609954651015044133532162580497588790301107135010149317318263331341596668228945003736897973385217973649395433413594450761042651672205510295923061557768349855131929164385281362357908051858782920150329798887139178923115839712114170], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4.C_2^5', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 40, 'pgroup': 0, 'primary_abelian_invariants': [4, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 14], [3, 4], [4, 20], [6, 10], [8, 8], [12, 10], [16, 6], [24, 2], [32, 4], [48, 4]], 'representations': {'PC': {'code': '4374104001971812637210868657296110215910690691632974045662714270831884873597020723817377879952630871178771851970344968', 'gens': [1, 4, 9, 10, 11], 'pres': [11, -2, -2, -2, -2, -2, -2, -2, -3, -2, 2, 2, 22, 56, 1608, 134819, 42430, 124, 41364, 42695, 158, 149957, 64432, 192, 54214, 61617, 226, 191495, 90130, 19049, 11932, 4815, 3044, 1273, 232330, 29083, 2958, 7325, 802, 1902]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [32785, 98307, 67566, 444639, 294921, 124284, 33281, 557073, 573953, 713485, 828161]}, 'Perm': {'d': 40, 'gens': [106725278220396658682310360437065777294176856320, 85946506876778786839613506412246644828980518400, 43980516372435047537341729429260515801035532285, 21583860631528492200752571982553339407830281647, 127885534293755012324800517846834471011426579200, 10080, 106890825587930069984075720886007484472064190760, 106890825587930069984075720886007484472064185600, 193075178158260629138208243878660206787155200, 86167098932475055551957252829593389915556369960, 179311424803903420894810092865203991933591696]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 8], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_4\\times C_8).\\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 192, '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': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 3], 'aut_gens': [[1, 8], [1, 40], [3, 8], [3, 12], [27, 8], [5, 8], [17, 8]], 'aut_group': '96.209', 'aut_hash': 209, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 9, 'aut_perms': [5160, 120, 1, 136, 7, 45360], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 2, 1, 1], [4, 1, 2, 2], [6, 2, 1, 1], [6, 2, 2, 1], [8, 3, 8, 1], [12, 2, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.11', 'autcent_hash': 11, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 1], [4, 1, 4], [6, 2, 3], [8, 3, 8], [12, 2, 4]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['24.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 1, 2, 2], [6, 2, 1, 3], [8, 3, 4, 2], [12, 2, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 24, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '12.4', 'hash': 9, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 40], [17, 8]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 8]], 'label': '48.9', 'linC_count': 48, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 4, 'linQ_dim': 5, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6:C8', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 24, 'number_divisions': 14, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 22, 'number_subgroups': 28, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [8, 24], [12, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[25, 8], [1, 12], [7, 8], [5, 8]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [6, 289, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 4]], 'representations': {'PC': {'code': 2900468177249, 'gens': [1, 4], 'pres': [5, -2, -2, -2, -2, -3, 10, 26, 803, 58, 804]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [706304316034, 140954895519]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [106892, 1692709, 518956, 1449557, 1565004]}, 'Perm': {'d': 13, 'gens': [39923583, 375196, 18550, 12316, 518918400]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6:C_8', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
