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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1728.3085', 'ambient_counter': 3085, 'ambient_order': 1728, 'ambient_tex': 'C_{12}^2.D_6', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 24, 'counter': 150, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.3085.36.i1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '36.i1.a1', 'outer_equivalence': False, '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': 36, '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': '48.23', 'subgroup_hash': 23, 'subgroup_order': 48, 'subgroup_tex': 'C_2\\times C_{24}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.3085', 'aut_centralizer_order': 288, 'aut_label': '36.i1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '16.14', 'aut_weyl_index': 2592, 'centralizer': '36.i1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.k1.a1', '18.b1.a1', '18.d1.a1', '18.f1.a1'], 'contains': ['72.a1.a1', '72.s1.a1', '108.d1.a1'], 'core': '72.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [9927, -1, 1219, -1, 570, -1, 9642, -1], 'generators': [6, 24, 888, 16, 12], 'label': '1728.3085.36.i1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '36.i1.a1', 'projective_image': '216.101', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36.i1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '48.23', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 4], 'aut_gens': [[1, 2], [1, 14], [1, 22], [25, 22], [25, 15]], 'aut_group': '32.46', 'aut_hash': 46, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [16, 23, 11536, 16576], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 2, 1], [4, 1, 2, 2], [6, 1, 2, 1], [6, 1, 4, 1], [8, 1, 8, 1], [12, 1, 4, 2], [24, 1, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.46', 'autcent_hash': 46, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times D_4', '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, 3], [3, 1, 2], [4, 1, 4], [6, 1, 6], [8, 1, 8], [12, 1, 8], [24, 1, 16]], 'center_label': '48.23', 'center_order': 48, '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', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 23, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [4, 1, 2, 2], [6, 1, 2, 3], [8, 1, 4, 2], [12, 1, 4, 2], [24, 1, 8, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 24, 'exponent': 24, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '12.5', 'hash': 23, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 48]], 'label': '48.23', 'linC_count': 384, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C24', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, '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': 12, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 48, 'number_divisions': 16, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 22, 'number_subgroups': 22, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [8, 8], [12, 8], [24, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 14], [1, 22], [25, 22], [25, 15]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [16, 23, 11536, 16576], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 8, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 4], [8, 2]], 'representations': {'PC': {'code': 1286603201, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -2, -3, 26, 42, 58]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [24992906222183252, 41624324273564377]}, 'GLFp': {'d': 2, 'p': 73, 'gens': [17894809, 20228936]}, 'Perm': {'d': 13, 'gens': [27760320, 479001600, 4, 11614560, 3669864]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 24], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{24}', 'transitive_degree': 48, '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': 18, 'aut_gen_orders': [6, 6, 6, 18, 6, 18, 2, 18, 18, 18], 'aut_gens': [[1, 2, 48], [889, 1474, 552], [897, 962, 1416], [33, 910, 240], [33, 226, 648], [33, 106, 1128], [33, 814, 1200], [25, 10, 912], [1, 814, 72], [17, 802, 336], [9, 686, 72]], 'aut_group': None, 'aut_hash': 5405983025374215726, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 41472, 'aut_permdeg': 88, 'aut_perms': [82546211689283535827084095443449590127052047141968592518691975903150985610929684645423799258727334839602969008325829644091400978133183, 86138638090379793250762440828260384967376401572803095931976570572700601572336747889827235329732162737313134147478586504814480080121715, 181706011251215555974778480444838187374751374698391301131808437371452266944451526647263538210664395380950389361320585420848270500482584, 84752248640800329250438245243927889877056344757746602554295178728954361408978410616986910245509555373755979551400232258020897483382505, 88980771891921905756489314406716381366269331641828686430631868999450160458682269561075765912901693579399128224300283128873700445567969, 103733386887883165812919960851710667990900414791416482498365920352743829347414055196309159936081484571022732467040011704442589781390074, 98084388415265320827036375708097444298220510431964331031644632047252185847314686307687652185134908704168867343247993908752992557635752, 80122388780869054551228111840460023544788584774084293996893670567690170158248088031470274343575443123622610279748965153641179460397296, 84524097747705874701023696073967702567086817465310404187559665452742308754580682285430026274416942636377957686737517090502572897027718, 182790228898179791441121304472045822201057434188939188503316464412395242613211189849963887178293156605003906994576874962077477956390542], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 2], [3, 4, 1, 1], [4, 1, 2, 2], [4, 4, 1, 2], [4, 6, 2, 2], [4, 12, 1, 2], [6, 2, 1, 6], [6, 4, 1, 3], [8, 18, 4, 1], [8, 36, 2, 1], [8, 54, 4, 1], [8, 108, 2, 1], [9, 2, 3, 1], [9, 4, 3, 1], [12, 2, 2, 4], [12, 4, 2, 6], [12, 4, 4, 2], [12, 12, 2, 4], [18, 2, 3, 3], [18, 4, 3, 3], [24, 36, 4, 2], [36, 2, 6, 2], [36, 4, 6, 4], [36, 4, 12, 2], [36, 12, 6, 4]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_9.(C_2\\times C_6^2).C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '324.118', 'autcentquo_hash': 118, 'autcentquo_nilpotent': False, 'autcentquo_order': 324, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2.S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 2], [3, 4, 1], [4, 1, 4], [4, 4, 2], [4, 6, 4], [4, 12, 2], [6, 2, 6], [6, 4, 3], [8, 18, 4], [8, 36, 2], [8, 54, 4], [8, 108, 2], [9, 2, 3], [9, 4, 3], [12, 2, 8], [12, 4, 20], [12, 12, 8], [18, 2, 9], [18, 4, 9], [24, 36, 8], [36, 2, 12], [36, 4, 48], [36, 12, 24]], 'center_label': '8.2', 'center_order': 8, 'central_product': False, 'central_quotient': '216.101', 'commutator_count': 1, 'commutator_label': '108.29', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 3085, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 2], [3, 4, 1, 1], [4, 1, 2, 2], [4, 4, 1, 2], [4, 6, 1, 2], [4, 6, 2, 1], [4, 12, 1, 2], [6, 2, 1, 6], [6, 4, 1, 3], [8, 18, 4, 1], [8, 36, 2, 1], [8, 54, 2, 2], [8, 108, 2, 1], [9, 2, 3, 1], [9, 4, 3, 1], [12, 2, 2, 4], [12, 4, 2, 6], [12, 4, 4, 2], [12, 12, 1, 2], [12, 12, 2, 3], [18, 2, 3, 3], [18, 4, 3, 3], [24, 36, 4, 2], [36, 2, 6, 2], [36, 4, 6, 4], [36, 4, 12, 2], [36, 12, 3, 2], [36, 12, 6, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 32256, 'exponent': 72, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '24.9', 'frattini_quotient': '72.46', 'hash': 3085, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 6, 18], 'inner_gens': [[1, 34, 936], [17, 2, 1680], [889, 98, 48]], 'inner_hash': 101, 'inner_nilpotent': False, 'inner_order': 216, 'inner_split': True, 'inner_tex': 'S_3\\times D_{18}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 92], [4, 84]], 'label': '1728.3085', 'linC_count': 192, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 160, 'linQ_dim': 16, 'linQ_dim_count': 96, 'linR_count': 60, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C12^2.D6', 'ngens': 9, '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 66, 'number_characteristic_subgroups': 113, 'number_conjugacy_classes': 192, 'number_divisions': 70, 'number_normal_subgroups': 121, 'number_subgroup_autclasses': 300, 'number_subgroup_classes': 322, 'number_subgroups': 1400, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 3], [3, 8], [4, 60], [6, 24], [8, 576], [9, 18], [12, 192], [18, 54], [24, 288], [36, 504]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 34, 48], [1, 34, 72], [25, 34, 72], [1, 10, 72], [889, 2, 48], [1, 22, 1392]], 'outer_group': '192.1543', 'outer_hash': 1543, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 15, 'outer_perms': [87178291200, 24, 479001624, 3628824, 40320, 747], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [4, 13], [6, 8], [8, 7], [12, 6], [16, 3], [24, 5], [48, 2]], 'representations': {'PC': {'code': 33037093201411022066312817612920423874923559301541931967064301075, 'gens': [1, 2, 6], 'pres': [9, 2, 2, 2, 2, 3, 2, 2, 3, 3, 7992, 613, 46, 542, 74, 1443, 102, 1444, 50549, 45374, 158, 51423, 186, 55312, 286, 46673]}, 'Perm': {'d': 28, 'gens': [357008514878953, 868708690478938093719080452, 12933424655, 4297243488701, 21821, 1321086775680, 6758061133824000, 12145280938112229701713920000, 23454917710790789398855680000]}}, 'schur_multiplier': [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_{12}^2.D_6', 'transitive_degree': 576, 'wreath_data': None, 'wreath_product': False}
