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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '768.1088764', 'ambient_counter': 1088764, 'ambient_order': 768, 'ambient_tex': 'D_4\\times \\GL(2,\\mathbb{Z}/4)', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 32, 'counter': 204, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '768.1088764.12.dv1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '12.dv1', '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': 12, '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': '64.67', 'subgroup_hash': 67, 'subgroup_order': 64, 'subgroup_tex': 'C_2^3.D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '768.1088764', 'aut_centralizer_order': None, 'aut_label': '12.dv1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '96.j1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['4.s1', '6.k1', '6.n1', '6.o1'], 'contains': ['24.f1', '24.dt1', '24.dv1', '24.eg1', '24.eo1', '24.eq1', '24.et1'], 'core': '24.f1', 'coset_action_label': None, 'count': 6, 'diagramx': None, 'generators': [2711, 5601, 12967], 'label': '768.1088764.12.dv1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.s1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '12.dv1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.dv1', 'subgroup_fusion': None, 'weyl_group': None}
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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': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[543, 533, 2597, 3981], [1549, 2609, 2597, 2719], [543, 3645, 1539, 667], [3593, 3603, 2597, 667], [2587, 565, 2597, 3981], [543, 3635, 2597, 1929], [543, 533, 2597, 4013], [1581, 1591, 2597, 1961], [3593, 3635, 2597, 2719], [543, 3635, 2597, 2719], [2619, 565, 2597, 2751]], 'aut_group': '1024.djl', 'aut_hash': 7296783744228449167, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 1024, 'aut_permdeg': 24, 'aut_perms': [3989969729018243082341, 25808031963714402771689, 206924722185558510021, 330898858647425103716894, 322005249717866456388570, 486534016350034979189760, 101826243778772913741994, 318633490761978528137563, 396031300260399829019670, 95490359517754471740443], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 2, 2], [2, 2, 4, 1], [2, 4, 2, 1], [4, 2, 8, 1], [4, 4, 2, 1], [4, 4, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9.C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '512.10494213', 'autcent_hash': 1718285292446712970, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^9', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 4], [2, 4, 2], [4, 2, 8], [4, 4, 6]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '8.5', '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'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 67, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 4], [2, 4, 1, 2], [4, 2, 2, 4], [4, 4, 1, 2], [4, 4, 2, 2]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 84, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '8.5', 'hash': 67, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 2], 'inner_gens': [[543, 533, 2597, 667], [543, 533, 2597, 1929], [543, 533, 2597, 3981], [3593, 2609, 2597, 3981]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12]], 'label': '64.67', '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': 'C2^3.D4', 'ngens': 3, 'nilpotency_class': 2, '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': 11, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 28, 'number_divisions': 22, 'number_normal_subgroups': 53, 'number_subgroup_autclasses': 66, 'number_subgroup_classes': 143, 'number_subgroups': 249, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 23], [4, 40]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [513, 513, 513, 513, 513, 513], 'outer_gens': [[2587, 2609, 2597, 3981], [543, 533, 2597, 4013], [575, 3635, 2597, 3981], [1581, 1559, 2597, 4013], [543, 565, 2597, 3981], [2619, 2607, 1539, 4013]], 'outer_group': '128.2163', 'outer_hash': 2163, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [5160, 11520, 40279703, 127387583, 7, 87465617], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 2]], 'representations': {'PC': {'code': 1103882096914, 'gens': [1, 2, 4, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 31, 1444, 370, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101736067518008, 91655858515233200, 125101729084381808]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [12967, 18688, 5601, 8645, 2665, 1801]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [643, 2565, 523, 545, 659, 1539]}, 'Perm': {'d': 12, 'gens': [55285327, 95242440, 139725365, 167368440, 16, 223347720]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_2^3.D_4', 'transitive_degree': 32, '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': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 4, 4, 6, 4, 6, 6, 2, 12], 'aut_gens': [[1733, 2671, 2707, 5922, 12967, 12175], [12107, 2671, 12601, 5394, 2665, 12967], [5811, 13033, 9155, 807, 12175, 12967], [6435, 13033, 17830, 1605, 12967, 2665], [19015, 13033, 2239, 6252, 2665, 12967], [16809, 13033, 9155, 1605, 12175, 12967], [5811, 2671, 19517, 16620, 2665, 12967], [1733, 19949, 13465, 6780, 2665, 12967], [1733, 2671, 2239, 10575, 12175, 12967], [6435, 9587, 12601, 5922, 2665, 12967]], 'aut_group': None, 'aut_hash': 1694722777932669743, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 36, 'aut_perms': [319779766416827653616641420083485802644640, 212980271589184922084851903942590410342543, 307145046823634039583972889993908647342283, 178810751301552218424035525181345488291387, 193434404152888823857033753329939122280285, 350519056774966131361470265833715782418932, 350684138360035321658046452568112556817000, 243295797354733915153967555162375513418097, 108127920886564277994955475515505991121905], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 2, 4, 1], [2, 3, 1, 4], [2, 4, 2, 1], [2, 6, 2, 1], [2, 6, 4, 1], [2, 12, 2, 2], [2, 24, 2, 1], [3, 8, 1, 1], [4, 2, 2, 1], [4, 4, 1, 1], [4, 6, 2, 1], [4, 12, 1, 1], [4, 12, 2, 3], [4, 24, 1, 4], [4, 24, 2, 3], [6, 8, 1, 3], [6, 8, 4, 1], [6, 16, 4, 2], [12, 16, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'A_4.C_2^6.C_2^4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '48.48', 'autcentquo_hash': 48, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 6], [2, 3, 4], [2, 4, 2], [2, 6, 6], [2, 12, 4], [2, 24, 2], [3, 8, 1], [4, 2, 2], [4, 4, 1], [4, 6, 2], [4, 12, 7], [4, 24, 10], [6, 8, 7], [6, 16, 8], [12, 16, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '192.1537', 'commutator_count': 1, 'commutator_label': '48.49', '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', '2.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1088764, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['8.3', 1], ['96.195', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 6], [2, 3, 1, 4], [2, 4, 1, 2], [2, 6, 1, 6], [2, 12, 1, 4], [2, 24, 1, 2], [3, 8, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 6, 1, 2], [4, 12, 1, 7], [4, 24, 1, 10], [6, 8, 1, 3], [6, 8, 2, 2], [6, 16, 1, 4], [6, 16, 2, 2], [12, 16, 1, 2], [12, 16, 2, 1]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 8255520, 'exponent': 12, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '192.1537', 'hash': 1088764, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 2, 6, 2, 2], 'inner_gens': [[1733, 2671, 2707, 5394, 12967, 12175], [1733, 2671, 12997, 5922, 12967, 12175], [1733, 13033, 2707, 12039, 2665, 12175], [8641, 2671, 7456, 5922, 12175, 2665], [1733, 2671, 13069, 17148, 12967, 12175], [1733, 2671, 2707, 16356, 12967, 12175]], 'inner_hash': 1537, 'inner_nilpotent': False, 'inner_order': 192, 'inner_split': True, 'inner_tex': 'C_2^3\\times S_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 24], [3, 16], [4, 5], [6, 8], [12, 1]], 'label': '768.1088764', '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': False, 'metacyclic': False, 'monomial': True, 'name': 'D4*GL(2,Z/4)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 39, 'number_characteristic_subgroups': 55, 'number_conjugacy_classes': 70, 'number_divisions': 65, 'number_normal_subgroups': 155, 'number_subgroup_autclasses': 1173, 'number_subgroup_classes': 3057, 'number_subgroups': 19814, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 167], [3, 8], [4, 344], [6, 184], [12, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [1729, 1729, 1729, 1729, 1729], 'outer_gens': [[12107, 9587, 2707, 5922, 12967, 12175], [1733, 2671, 9623, 12039, 2665, 12175], [12107, 9587, 9623, 5922, 12967, 12175], [12107, 9587, 9623, 17226, 12967, 12175], [16185, 9587, 9623, 17226, 12967, 12175]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [806400, 1174320, 127, 1174446, 368062], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [3, 16], [4, 7], [6, 8], [8, 1], [12, 1]], 'representations': {'PC': {'code': '448255217457503097348806986207539605152256557068617541636', 'gens': [1, 2, 3, 5, 8, 9], 'pres': [9, -2, -2, -2, -2, -2, -2, -3, -2, 2, 173, 74, 5044, 922, 130, 2183, 158, 2040, 10393, 1771, 1348, 493, 2960, 539, 791]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [5601, 19949, 1765, 8645, 18685, 2665, 1733, 1801, 9587]}, 'Perm': {'d': 12, 'gens': [1565, 5160, 40279680, 7632000, 87091200, 11520, 3, 7, 16]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_4\\times \\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
