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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '672.1102', 'ambient_counter': 1102, 'ambient_order': 672, 'ambient_tex': 'C_2^4.F_7', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 16, 'counter': 21, 'cyclic': False, 'direct': None, 'hall': 2, 'label': '672.1102.21.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '21.a1', 'outer_equivalence': True, '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': 21, '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': '32.45', 'subgroup_hash': 45, 'subgroup_order': 32, 'subgroup_tex': 'C_2^3\\times C_4', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '672.1102', 'aut_centralizer_order': 6, 'aut_label': '21.a1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '21504.y', 'aut_weyl_index': 42, 'centralizer': '7.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1', '7.a1'], 'contains': ['42.a1', '42.b1'], 'core': '42.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [2472, -1, 3831, -1], 'generators': [1, 2, 36, 336], 'label': '672.1102.21.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '3.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1', 'old_label': '21.a1', 'projective_image': '42.1', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '21.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 168, 'aut_gen_orders': [7, 3], 'aut_gens': [[1, 2, 4, 8], [20, 19, 2, 14], [1, 22, 19, 28]], 'aut_group': '21504.y', 'aut_hash': 2095861522221326572, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21504, 'aut_permdeg': 16, 'aut_perms': [19981329134312, 7769561399125], 'aut_phi_ratio': 1344.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [4, 1, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:C_2^3:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': '21504.y', 'autcent_hash': 2095861522221326572, 'autcent_nilpotent': False, 'autcent_order': 21504, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4:C_2^3:\\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]], 'center_label': '32.45', 'center_order': 32, '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'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 45, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [4, 1, 2, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 15, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '16.14', 'hash': 45, '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, 32]], 'label': '32.45', 'linC_count': 13440, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 1792, 'linQ_dim': 5, 'linQ_dim_count': 1792, 'linR_count': 1792, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3*C4', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 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': 32, 'number_divisions': 24, 'number_normal_subgroups': 118, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 118, 'number_subgroups': 118, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15], [4, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 168, 'outer_gen_orders': [7, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[20, 19, 2, 14], [1, 22, 19, 28]], 'outer_group': '21504.y', 'outer_hash': 2095861522221326572, 'outer_nilpotent': False, 'outer_order': 21504, 'outer_permdeg': 16, 'outer_perms': [19981329134312, 7769561399125], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:C_2^3:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8]], 'representations': {'PC': {'code': 16392, 'gens': [1, 2, 3, 4], 'pres': [5, -2, 2, 2, 2, -2, 58]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 705686952884, 706461792892, 706460730980]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 122297461254, 30594431879, 30704310001, 30590220001]}, 'Perm': {'d': 10, 'gens': [22, 362880, 5040, 120, 7]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], '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_4', 'transitive_degree': 32, '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': '96.220', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [2, 2, 6, 2, 2, 2, 2, 2, 2, 7], 'aut_gens': [[1, 2, 4, 48], [1, 3, 4, 625], [1, 2, 6, 624], [1, 2, 4, 144], [1, 2, 29, 624], [1, 2, 342, 624], [1, 26, 4, 648], [339, 2, 342, 624], [1, 2, 28, 624], [1, 2, 4, 626], [1, 2, 196, 48]], 'aut_group': '903168.c', 'aut_hash': 4587490310355808623, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 903168, 'aut_permdeg': 119, 'aut_perms': [41104250492830006968853304107527545435967170295140373398323340843902747884880132403266447024290478485639914415634686438920202301411839185834587582761487571435141380472370506316320813283019772870094, 37325246780205510577717438555484902744049775808389814363358708512019327575215305605676616432415362305384261097296664028339472396753773446846480454522040973290637321961273143378068325945929350614048, 14173257510759373042688839959407615527895509047089056565031343053780733205389708208821487132055695692259226331243116229083880110242580737986518484453428207522801804812146514847078977798476576435504, 29759128055985794141390809671974900195952003333367352142460957442856590469766922965252352539380348939043746368818351706244394106640654279606949529500239891711141808880663692044071997814924723649279, 28345522705643533497337036995415743302900009307376091639081081104246066964647010840259202163909696201202763355936793444798884779461448787277474462694758927621382870174771637277583638012576045346560, 41104250492829960731383122603171213283548924216142017113189753830964455776210275271440182048215776292585323003293051719660650150119508589443971194343401466381614528592815471385617824597426328127296, 28452739382399923867616932380431426112179138505662046526648604098316292047848095749910979706439852744448405920430401734520641413422570664681617199020031947379831005126918212345959185983917122975895, 29290641539901206866167819001818718835134174050181518054525497431034706663483677074867788712414617460939148837834570995992797301875437179065414462585366017213970256078486685641456708480878025777232, 37324870315560038560965166306180113996216400054137895663946583772402818741822305411666151052370657170900912908501706449616280315738518518395456493133555644347602164792341813249776495813669535266848, 7086851268010977346172878911836603810275398959887446034392684713610623156625235542892901101696907438034861418218166309333629429165629529707971943762667502776877212995826567167376948037126457088576], 'aut_phi_ratio': 4704.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [3, 7, 1, 2], [4, 7, 16, 1], [6, 7, 1, 2], [6, 7, 14, 2], [7, 6, 1, 1], [12, 7, 16, 2], [14, 6, 1, 1], [14, 6, 14, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_7\\times C_2^4:C_2^3:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': '21504.y', 'autcent_hash': 2095861522221326572, 'autcent_nilpotent': False, 'autcent_order': 21504, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4:C_2^3:\\GL(3,2)', '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, 15], [3, 7, 2], [4, 7, 16], [6, 7, 30], [7, 6, 1], [12, 7, 32], [14, 6, 15]], 'center_label': '16.14', 'center_order': 16, 'central_product': True, 'central_quotient': '42.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1102, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['84.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [3, 7, 2, 1], [4, 7, 2, 8], [6, 7, 2, 15], [7, 6, 1, 1], [12, 7, 4, 8], [14, 6, 1, 15]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 68400, 'exponent': 84, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '336.216', 'hash': 1102, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [1, 1, 6, 7], 'inner_gens': [[1, 2, 4, 48], [1, 2, 4, 48], [1, 2, 4, 240], [1, 2, 484, 48]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 42, 'inner_split': True, 'inner_tex': 'F_7', 'inner_used': [3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 96], [6, 16]], 'label': '672.1102', 'linC_count': None, 'linC_degree': None, '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': 'C2^4.F7', '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, 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': 11, 'number_conjugacy_classes': 112, 'number_divisions': 64, 'number_normal_subgroups': 303, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 472, 'number_subgroups': 1486, 'old_label': None, 'order': 672, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 15], [3, 14], [4, 112], [6, 210], [7, 6], [12, 224], [14, 90]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [7, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[362, 3, 5, 385], [361, 362, 4, 409]], 'outer_group': '21504.y', 'outer_hash': 2095861522221326572, 'outer_nilpotent': False, 'outer_order': 21504, 'outer_permdeg': 16, 'outer_perms': [20448581743843, 791677760034], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:C_2^3:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 24], [4, 8], [6, 16]], 'representations': {'PC': {'code': 1571821113398409421955241546966889001, 'gens': [1, 2, 3, 6], 'pres': [7, -2, -2, -2, -2, -3, -2, -7, 58, 80, 2539, 2798, 1167, 124, 5900, 2379, 622]}, 'GLZN': {'d': 2, 'p': 56, 'gens': [1580569, 177185, 2634255, 265117, 177213, 7200297, 176065]}, 'Perm': {'d': 17, 'gens': [1321086821047, 136, 127, 806416, 2889377568000, 806400, 24865491974400]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4.F_7', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
