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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '128.1613', 'ambient_counter': 1613, 'ambient_order': 128, 'ambient_tex': 'C_2^5:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 1, 'counter': 84, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '128.1613.32.s1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '32.s1', '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': 32, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '128.1613', 'aut_centralizer_order': 256, 'aut_label': '32.s1', 'aut_quo_index': None, 'aut_stab_index': 48, 'aut_weyl_group': '2.1', 'aut_weyl_index': 12288, 'centralizer': '4.e1', 'complements': None, 'conjugacy_class_count': 12, 'contained_in': ['16.p1', '16.q1', '16.x1'], 'contains': ['64.e1', '64.g1'], 'core': '128.a1', 'coset_action_label': '32T1293', 'count': 48, 'diagramx': [8155, -1, 8176, -1], 'generators': [90882847, 14928503], 'label': '128.1613.32.s1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.d1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.e1', 'old_label': '32.s1', 'projective_image': '128.1613', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.s1', '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2], 'aut_gens': [[41489407, 90882847, 136095256, 23, 14928480, 1871400, 136095240], [211614480, 302376367, 16, 136095263, 14928480, 1871400, 136095240], [41489416, 90882847, 136095256, 23, 14928480, 1871400, 136095240], [47985967, 90882847, 136095256, 23, 125199480, 134996400, 136095240], [211614503, 90882847, 136095256, 136095247, 14928480, 1871400, 136095240], [211614480, 302376360, 136095256, 23, 14928480, 1871400, 136095240], [211614480, 302376360, 23, 136095256, 14928480, 1871400, 136095240], [261768256, 90882847, 136095256, 23, 14928480, 16420440, 136095240], [41489416, 90882856, 136095256, 23, 14928480, 1871400, 136095240], [211614480, 90882847, 136095256, 23, 14928480, 1871400, 136095240], [200672767, 90882847, 136095256, 23, 14928480, 134996400, 136095240], [200672767, 98155447, 136095256, 23, 125199480, 134996400, 136095240], [211614496, 302376367, 16, 23, 14928480, 1871400, 136095240]], 'aut_group': '24576.bdu', 'aut_hash': 553978189397296999, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24576, 'aut_permdeg': 64, 'aut_perms': [15773264555839302740223895821665299178158210279594009403766865944741259575629642597376307, 119346503334918360444340409622394158217418822282526050837977794397112653613820718664290130, 65855955689047774246800900089652150350372081016462960922977410667907695277499496865860492, 1763130614654212034397955526933940717373872404533311251747811788324485142407298618770104, 1278103986315958628649266169681054881583425689424750630426670395029426202916760728829743, 199844765901076334906544019203373515465720872184884604181145023295466095914677838088, 12435317550878155256702823740356460523128053972078884020544180366796220191762706605701116, 357860985000180770323463527746323385506838792935270766064927572642378279010632113412061, 94492886217981469566303873492311786708597946125325328610770971829901725185868462294343050, 21245419826783460116514188442247260436593124559696549180674301146406576213055426784635597, 140830732455303245516327011620663847041136436451997396443336419901511379263355445487360, 67811484813635303858645803623251212959892042949990757026210380306642218768096992075911114], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 2, 1, 1], [2, 2, 2, 1], [2, 2, 3, 1], [2, 2, 6, 1], [2, 4, 4, 1], [4, 4, 4, 1], [4, 4, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^7.(D_4\\times S_4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': None, 'autcent_hash': 4599629986281706675, 'autcent_nilpotent': False, 'autcent_order': 1536, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4.C_2^4.S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': None, 'autcentquo_hash': 11, 'autcentquo_nilpotent': True, 'autcentquo_order': 16, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 12], [2, 4, 4], [4, 4, 20]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '16.3', 'commutator_count': 1, 'commutator_label': '4.2', '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': 1613, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['32.6', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 12], [2, 4, 1, 4], [4, 4, 1, 4], [4, 4, 2, 8]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 3360, 'exponent': 4, 'exponents_of_order': [7], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '16.14', 'hash': 1613, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 2, 1, 1, 2, 2, 1], 'inner_gens': [[41489407, 98155447, 136095256, 23, 125199480, 1871400, 136095240], [200672767, 90882847, 136095256, 23, 14928480, 134996400, 136095240], [41489407, 90882847, 136095256, 23, 14928480, 1871400, 136095240], [41489407, 90882847, 136095256, 23, 14928480, 1871400, 136095240], [211614487, 90882847, 136095256, 23, 14928480, 1871400, 136095240], [41489407, 302376367, 136095256, 23, 14928480, 1871400, 136095240], [41489407, 90882847, 136095256, 23, 14928480, 1871400, 136095240]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': False, 'inner_tex': 'C_2^2:C_4', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8], [4, 4]], 'label': '128.1613', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'C2^5:C4', 'ngens': 7, 'nilpotency_class': 3, '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': 10, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 44, 'number_divisions': 36, 'number_normal_subgroups': 180, 'number_subgroup_autclasses': 93, 'number_subgroup_classes': 556, 'number_subgroups': 1228, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 47], [4, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[41489407, 90882847, 136095247, 23, 14928480, 1871400, 136095240], [41489407, 90882847, 136095247, 16, 14928480, 1871400, 136095240], [40739047, 90882847, 136095256, 23, 125199480, 1871400, 136095240], [168135960, 90882847, 136095256, 23, 14928480, 16420440, 136095240], [41489407, 90882847, 16, 136095263, 14928480, 1871400, 136095240], [41489407, 90882847, 16, 23, 14928480, 1871400, 136095240], [41489407, 90882863, 136095256, 23, 14928480, 1871400, 136095240], [41489407, 90882856, 136095256, 23, 14928480, 1871400, 136095240], [41489400, 90882847, 136095256, 23, 14928480, 1871400, 136095240], [41489416, 90882847, 136095256, 23, 14928480, 1871400, 136095240]], 'outer_group': '1536.408639927', 'outer_hash': 1736637719712388863, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 14, 'outer_perms': [6442294375, 79914421, 316, 6402372480, 314, 82, 14377673789, 21196264402, 14856750720, 26951102640], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 4]], 'representations': {'PC': {'code': 37438389184364580, 'gens': [1, 2, 3, 5, 6], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 58, 718, 1195, 530, 124]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101740294620861, 125101724450770357, 25038659807093174, 24992906222183252]}, 'Perm': {'d': 12, 'gens': [41489407, 90882847, 136095256, 23, 14928480, 1871400, 136095240]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^5:C_4', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
