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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '384.12649', 'ambient_counter': 12649, 'ambient_order': 384, 'ambient_tex': '(C_2\\times D_8).D_6', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 4, 'counter': 314, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '384.12649.48.i1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '48.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': 48, '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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.12649', 'aut_centralizer_order': 768, 'aut_label': '48.i1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '4.2', 'aut_weyl_index': 1536, 'centralizer': '4.bo1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['16.m1.a1', '24.d1.a1', '24.h1.a1', '24.n1.a1', '24.bl1.a1', '24.bm1.a1', '24.bn1.a1', '24.bo1.a1'], 'contains': ['96.a1.a1', '96.y1.a1', '96.y1.b1'], 'core': '96.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [5577, -1, 5712, -1, 5092, -1, 5340, -1], 'generators': [97, 16], 'label': '384.12649.48.i1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '24.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.c1.a1', 'old_label': '48.i1.a1', 'projective_image': '96.209', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '48.i1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': '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], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', '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': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], '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_4', 'transitive_degree': 8, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [6, 2, 2, 6, 2, 6, 12, 2, 4, 4, 4], 'aut_gens': [[1, 2, 8, 32], [197, 150, 152, 32], [1, 214, 24, 160], [213, 6, 216, 352], [17, 326, 344, 224], [17, 18, 24, 352], [1, 258, 264, 224], [197, 274, 88, 240], [21, 262, 264, 368], [5, 274, 280, 368], [21, 18, 24, 160], [197, 130, 136, 160]], 'aut_group': None, 'aut_hash': 2404398907887337476, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6144, 'aut_permdeg': 32, 'aut_perms': [107400332930638245547288907273859869, 254814981871591429527738887712200590, 55993481710168583550453260055243132, 224644768573282453158887631347459916, 56842590909426571513139263218370362, 24947838943621823564561234470629128, 92992419202263410249498083630849893, 37743355932503921498048739112195004, 84768089566498190775013027688985573, 50087178648002808011859320747597544, 172662877125183369407172731628275650], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 4, 2, 1], [2, 8, 1, 1], [2, 24, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 4, 2, 1], [4, 6, 2, 2], [4, 8, 1, 1], [4, 12, 1, 2], [4, 12, 2, 2], [4, 24, 1, 1], [6, 2, 1, 3], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 1, 1], [8, 4, 2, 1], [8, 8, 1, 1], [8, 12, 2, 1], [8, 24, 1, 1], [12, 4, 1, 2], [12, 4, 2, 1], [12, 8, 2, 1], [12, 16, 1, 1], [24, 8, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^6.C_2^5)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '24.14', 'autcentquo_hash': 14, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 3], [2, 8, 1], [2, 24, 1], [3, 2, 1], [4, 2, 2], [4, 4, 3], [4, 6, 4], [4, 8, 1], [4, 12, 6], [4, 24, 1], [6, 2, 3], [6, 8, 3], [6, 16, 1], [8, 4, 2], [8, 8, 1], [8, 12, 2], [8, 24, 1], [12, 4, 4], [12, 8, 2], [12, 16, 1], [24, 8, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '96.209', 'commutator_count': 1, 'commutator_label': '24.9', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 12649, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 3], [2, 8, 1, 1], [2, 24, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 3], [4, 6, 2, 2], [4, 8, 1, 1], [4, 12, 1, 2], [4, 12, 2, 2], [4, 24, 1, 1], [6, 2, 1, 3], [6, 8, 1, 3], [6, 16, 1, 1], [8, 4, 2, 1], [8, 8, 1, 1], [8, 12, 2, 1], [8, 24, 1, 1], [12, 4, 1, 2], [12, 4, 2, 1], [12, 8, 1, 2], [12, 16, 1, 1], [24, 8, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1048320, 'exponent': 24, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '48.51', 'hash': 12649, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 2, 6], 'inner_gens': [[1, 198, 8, 32], [5, 2, 8, 176], [1, 2, 8, 352], [1, 274, 72, 32]], 'inner_hash': 209, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': None, 'inner_tex': 'D_4\\times D_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 20], [4, 14], [8, 1]], 'label': '384.12649', 'linC_count': 32, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 16, 'linQ_dim': 14, 'linQ_dim_count': 64, 'linR_count': 64, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*D8).D6', 'ngens': 8, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 38, 'number_characteristic_subgroups': 109, 'number_conjugacy_classes': 51, 'number_divisions': 42, 'number_normal_subgroups': 117, 'number_subgroup_autclasses': 362, 'number_subgroup_classes': 396, 'number_subgroups': 1294, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 47], [3, 2], [4, 144], [6, 46], [8, 64], [12, 48], [24, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2, 2], 'outer_gen_pows': [128, 0, 0, 0, 260, 0], 'outer_gens': [[197, 150, 152, 32], [1, 214, 24, 160], [213, 6, 216, 352], [17, 18, 24, 352], [197, 274, 88, 240], [21, 262, 264, 368]], 'outer_group': '64.267', 'outer_hash': 267, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 64, 'outer_perms': [1987252695887700205215063048327198338549971052505810857474097390497050149140808214496830, 4280000852525512180267781122454722320080274048618517776111857050091116638327077994254444, 6299579328022735326339682448497438100391067604092077975988571528938477382077949002022685, 10330447947115170887725080862647977022069724169167492944174333798385272639576728874368638, 14359639308393299087352881514633755063104226983857490440363110941234062819738085294080000, 16122473818012755367313102789488863604121271888846132947015757830740667827110195088971150], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 10], [8, 3], [16, 1]], 'representations': {'PC': {'code': 90894005787658474817664513923373804810268161, 'gens': [1, 2, 4, 6], 'pres': [8, -2, -2, -2, -2, -2, 2, -2, -3, 3169, 41, 4706, 1170, 91, 4237, 2141, 141, 8974, 2270, 166, 8207, 2079]}, 'Perm': {'d': 27, 'gens': [405234858872999126641108345, 870677086695415830486407640, 78385920768906056731681464, 4370528679143168242322881, 8361600, 1307991117668345899499587200, 1726294691134515892270003200, 3]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times D_8).D_6', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
