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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '768.1089108', 'ambient_counter': 1089108, 'ambient_order': 768, 'ambient_tex': 'C_2^5:S_4', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 1, 'counter': 175, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '768.1089108.48.br2', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '48.br2', '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': 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': '16.11', 'subgroup_hash': 11, 'subgroup_order': 16, 'subgroup_tex': 'C_2\\times D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '768.1089108', 'aut_centralizer_order': 32, 'aut_label': '48.br2', 'aut_quo_index': None, 'aut_stab_index': 144, 'aut_weyl_group': '8.5', 'aut_weyl_index': 4608, 'centralizer': '48.bj1', 'complements': None, 'conjugacy_class_count': 12, 'contained_in': ['24.h1', '24.l2'], 'contains': ['96.x1', '96.y1', '96.ba2', '96.cf2', '96.cg1'], 'core': '768.a1', 'coset_action_label': None, 'count': 144, 'diagramx': [1588, -1, 2290, -1], 'generators': [138989527, 178981920, 135359400, 41909167], 'label': '768.1089108.48.br2', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '12.n1', 'old_label': '48.br2', 'projective_image': '768.1089108', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '48.br2', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', '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], 'aut_gens': [[126, 55, 289, 288], [127, 55, 289, 288], [126, 265, 1, 288], [54, 127, 289, 288], [414, 55, 289, 288], [127, 54, 289, 288], [414, 265, 289, 288]], 'aut_group': '64.138', 'aut_hash': 138, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 8, 'aut_perms': [2309, 526, 5329, 3043, 12316, 18498], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [4, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', '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, 3], [2, 2, 4], [4, 2, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [4, 2, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 21, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 1], 'inner_gens': [[126, 265, 289, 288], [414, 55, 289, 288], [126, 55, 289, 288], [126, 55, 289, 288]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.11', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 8, 'linQ_dim': 3, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 10, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 27, 'number_subgroups': 35, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 11], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 151, 0], 'outer_gens': [[54, 127, 289, 288], [127, 55, 289, 288], [415, 54, 1, 288], [127, 54, 289, 288]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2]], 'representations': {'PC': {'code': 8772, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 78, 34]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16322, 16432, 3198]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8912, 8156, 13286, 14044]}, 'Perm': {'d': 6, 'gens': [126, 55, 289, 288]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times D_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': 3, 'aut_exponent': 24, 'aut_gen_orders': [4, 6, 4, 4], 'aut_gens': [[41909167, 16, 7, 41546183, 404040, 94434600, 138989520, 142618440, 175351800], [41896, 7, 16, 41546176, 142329720, 124104960, 83502720, 142618440, 98064720], [83463960, 7, 23, 216080063, 123380040, 164062080, 83502720, 175351800, 98064720], [215717063, 23, 7, 216080040, 22584960, 94434600, 138989520, 142618440, 175351800], [174988807, 16, 23, 216080047, 135072720, 83502720, 164062080, 98064720, 175351800]], 'aut_group': None, 'aut_hash': 5262991014587272500, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 36864, 'aut_permdeg': 96, 'aut_perms': [772714307080193948596245314395470678987540429682134070805545551774838532476248331521791205051633109364917834119085241032197003108207183764940958189582, 793144268631617530539101509629531611899506247755729885937408166667025478047508112856000984226637485485740146107815997592996341328207178354378179848577, 196598133803143259856666673395712965564174191253328715621850068395323375961918543889467813661819347889236063005844198241158953815040531251563147154893, 618301449948346046995295882088871918497707562500374206053917610902294545466234026497973390953482036262427523402693295355483987457510498021278461949808], 'aut_phi_ratio': 144.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 1, 1], [2, 3, 3, 1], [2, 4, 4, 1], [2, 6, 2, 1], [2, 6, 6, 1], [2, 12, 8, 1], [3, 32, 1, 1], [4, 12, 4, 1], [4, 12, 8, 1], [4, 24, 8, 1], [6, 32, 3, 1], [6, 32, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^8.C_3.C_6.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '96.227', 'autcent_hash': 227, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^2:S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '384.5678', 'autcentquo_hash': 5678, 'autcentquo_nilpotent': False, 'autcentquo_order': 384, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^4:S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [2, 4, 4], [2, 6, 8], [2, 12, 8], [3, 32, 1], [4, 12, 12], [4, 24, 8], [6, 32, 7]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '192.955', 'commutator_count': 1, 'commutator_label': '48.50', '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': 1089108, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['192.955', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [2, 4, 1, 4], [2, 6, 1, 8], [2, 12, 1, 8], [3, 32, 1, 1], [4, 12, 1, 12], [4, 24, 1, 8], [6, 32, 1, 7]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2751840, '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': 1089108, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 1, 1, 2, 3, 2, 2, 2, 2], 'inner_gens': [[41909167, 16, 7, 41546183, 727200, 182610840, 138989520, 98064720, 175351800], [41909167, 16, 7, 41546183, 404040, 94434600, 138989520, 142618440, 175351800], [41909167, 16, 7, 41546183, 404040, 94434600, 138989520, 142618440, 175351800], [41909167, 16, 7, 41546183, 404040, 178981920, 101693640, 142618440, 175351800], [41505127, 16, 7, 41546183, 404040, 138989520, 182610840, 175351800, 98064720], [251894887, 16, 7, 291811823, 182171520, 94434600, 138989520, 142618440, 175351800], [41909167, 16, 7, 216080063, 95519640, 94434600, 138989520, 142618440, 175351800], [215717047, 16, 7, 41546183, 99146880, 94434600, 138989520, 142618440, 175351800], [41909167, 16, 7, 41546183, 141569280, 94434600, 138989520, 142618440, 175351800]], 'inner_hash': 955, 'inner_nilpotent': False, 'inner_order': 192, 'inner_split': True, 'inner_tex': 'C_2^3:S_4', 'inner_used': [1, 4, 5, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 8], [3, 16], [6, 16]], 'label': '768.1089108', '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': 'C2^5:S4', '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': 14, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 56, 'number_divisions': 56, 'number_normal_subgroups': 104, 'number_subgroup_autclasses': 336, 'number_subgroup_classes': 3002, 'number_subgroups': 20340, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 175], [3, 32], [4, 336], [6, 224]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[41909167, 16, 23, 41546167, 404040, 94434600, 138989520, 142618440, 175351800], [363007, 16, 7, 41546183, 404040, 22182600, 14923560, 142618440, 175351800], [41909167, 23, 16, 41546167, 404040, 94434600, 138989520, 142618440, 175351800], [41909167, 16, 7, 41546160, 404040, 94434600, 138989520, 142618440, 175351800], [41909167, 16, 7, 41546176, 404040, 94434600, 138989520, 142618440, 175351800], [41909160, 16, 7, 41546183, 404040, 178981920, 101693640, 142618440, 175351800], [41909183, 16, 7, 41546183, 404040, 94434600, 138989520, 142618440, 175351800]], 'outer_group': '192.955', 'outer_hash': 955, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 8, 'outer_perms': [121, 23616, 1444, 23, 16, 11536, 5167], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:S_4', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [3, 16], [6, 16]], 'representations': {'PC': {'code': '132797905744822825092045584005402144460714614061629195746609184', 'gens': [1, 2, 3, 5, 6, 7, 8], 'pres': [9, 2, 2, 2, 3, 2, 2, 2, 2, 2, 542, 74, 579, 11884, 3802, 2056, 1229, 30246, 7584, 3057, 5641, 1546, 2059, 916, 214, 1970, 4895]}, 'Perm': {'d': 12, 'gens': [41909167, 16, 7, 41546183, 404040, 94434600, 138989520, 142618440, 175351800]}}, 'schur_multiplier': [2, 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': 'C_2^5:S_4', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
