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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '4608.co', 'ambient_counter': 67, 'ambient_order': 4608, 'ambient_tex': 'C_2^2\\wr S_3:C_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 384, 'counter': 18, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4608.co.6.E', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.e1', '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': 6, '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': '768.1084399', 'subgroup_hash': 1084399, 'subgroup_order': 768, 'subgroup_tex': 'C_2^7:C_6', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4608.co', 'aut_centralizer_order': 2, 'aut_label': '6.E', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '3072.bhh', 'aut_weyl_index': 6, 'centralizer': '1152.B', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.A', '3.B'], 'contains': ['12.A', '12.K', '12.P', '12.Q', '12.R', '18.F', '24.M'], 'core': '12.A', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [8120557670400, 3152349481625, 12940663680, 194561740855, 12940669009, 31697930880, 454699889280, 12341, 1340852849280], 'label': '4608.co.6.E', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.A', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.B', 'old_label': '6.e1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.E', 'subgroup_fusion': None, 'weyl_group': '384.5747'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [4, 4, 8, 6, 6, 6], 'aut_gens': [[2003546160, 6438625326, 14377673521, 21196264608, 15335757414, 20717187409, 27430179840, 15335757360, 288], [20684200320, 32736493855, 27909186481, 21196264608, 15335757414, 27430180129, 20717187120, 15335757360, 288], [12973040640, 44602755247, 14856750721, 26951102928, 15335757624, 27430180129, 20717187120, 15335757360, 288], [15408252720, 44602755246, 2918004049, 2438997408, 27430179895, 20717187121, 15335757360, 27430179840, 288], [26904371760, 6438625326, 19239897601, 25470547488, 27430179895, 15335757649, 20717187120, 27430179840, 288], [14976057600, 32736493855, 26951102641, 14856751008, 20717187384, 27430180129, 15335757360, 20717187120, 288], [13016550960, 39452716926, 2918004049, 1959920208, 20717187384, 27430179841, 15335757360, 20717187120, 288]], 'aut_group': None, 'aut_hash': 7450110856054544156, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 52, 'aut_perms': [23524891142836394982862035191610114519582287022173360324871742659466, 44183245429891469545920418080202655169103474875538527360121306068930, 23989573519288901961457541623456781511545270969618751649422730794073, 57717173486992033662249470035079397463898383625409258173651511950116, 75249675612296491876278990623050344835913370157227903745411122292804, 65368314425270894773355192365048068838522264465058645049619646651050], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 6, 2, 3], [2, 6, 4, 1], [2, 6, 8, 1], [2, 8, 2, 1], [3, 16, 2, 1], [4, 8, 2, 1], [4, 24, 2, 2], [6, 16, 2, 1], [6, 16, 4, 1], [6, 32, 4, 2], [12, 32, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:C_3.C_2^4.C_2^4', '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': 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, 2, 2], [2, 3, 4], [2, 6, 18], [2, 8, 2], [3, 16, 2], [4, 8, 2], [4, 24, 4], [6, 16, 6], [6, 32, 8], [12, 32, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '192.1000', 'commutator_count': 1, 'commutator_label': '32.51', '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': 1084399, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['384.5790', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 3, 1, 4], [2, 6, 1, 18], [2, 8, 1, 2], [3, 16, 2, 1], [4, 8, 1, 2], [4, 24, 1, 4], [6, 16, 2, 3], [6, 32, 2, 4], [12, 32, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 10920, 'exponent': 12, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '96.228', 'hash': 1084399, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [3, 2, 2, 2, 2, 2, 2, 2, 1], 'inner_gens': [[2003546160, 32736493566, 21196264321, 27909186768, 20717187174, 27430180129, 15335757360, 20717187120, 288], [19160109360, 6438625326, 21675270961, 26951102928, 15335757624, 20717187409, 27430179840, 15335757360, 288], [27383287680, 39452716926, 14377673521, 21196264608, 15335757414, 20717187409, 27430179840, 15335757360, 288], [13016550960, 44602755246, 14377673521, 21196264608, 15335757414, 20717187409, 27430179840, 15335757360, 288], [27862329600, 6438625614, 14377673521, 21196264608, 15335757414, 20717187409, 27430179840, 15335757360, 288], [14976057600, 6438625326, 14377673521, 21196264608, 15335757414, 20717187409, 27430179840, 15335757360, 288], [19160109360, 6438625326, 14377673521, 21196264608, 15335757414, 20717187409, 27430179840, 15335757360, 288], [27862329600, 6438625326, 14377673521, 21196264608, 15335757414, 20717187409, 27430179840, 15335757360, 288], [2003546160, 6438625326, 14377673521, 21196264608, 15335757414, 20717187409, 27430179840, 15335757360, 288]], 'inner_hash': 1000, 'inner_nilpotent': False, 'inner_order': 192, 'inner_split': False, 'inner_tex': 'C_2^4:A_4', 'inner_used': [1, 2, 3, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 6], [3, 8], [6, 18]], 'label': '768.1084399', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^7:C6', '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': 22, 'number_characteristic_subgroups': 25, 'number_conjugacy_classes': 56, 'number_divisions': 46, 'number_normal_subgroups': 65, 'number_subgroup_autclasses': 802, 'number_subgroup_classes': 5670, 'number_subgroups': 33614, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 143], [3, 32], [4, 112], [6, 352], [12, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2], 'outer_gen_pows': [6438625561, 0, 0, 0, 79914240, 0], 'outer_gens': [[21642168240, 6438625327, 1959920209, 2918004048, 15335757415, 27430179841, 20717187120, 15335757360, 288], [2003546160, 6438625326, 14377673809, 21196264608, 15335757414, 20717187121, 27430179840, 15335757360, 288], [25390794240, 6438625327, 27909186481, 21196264608, 15335757415, 27430180129, 20717187120, 15335757360, 288], [2039788800, 6438625326, 26951102641, 21675271248, 27430179894, 20717187409, 15335757360, 27430179840, 288], [2482462080, 6438625327, 21196264321, 27909186768, 20717187384, 27430180129, 15335757360, 20717187120, 288], [2003546160, 6438625326, 14377673521, 21196264608, 15335757624, 20717187409, 27430179840, 15335757360, 288]], 'outer_group': '64.264', 'outer_hash': 264, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [2799055, 848928, 1956601, 1270441, 374407, 374406], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4:C_2^3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [3, 8], [4, 2], [6, 18]], 'representations': {'PC': {'code': '467529307210114767111331545093557958255045607323909203541001757788800', 'gens': [1, 3, 5, 6, 7, 8, 9], 'pres': [9, -2, -3, -2, -2, 2, 2, 2, 2, 2, 18, 11504, 3980, 74, 16204, 17023, 2362, 1949, 10058, 6719, 9078, 5685, 2040, 39751, 22912, 1177, 35000, 18971, 9098]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [1074840641, 2155454277, 1074846310, 3082718614, 434625224, 1074848632, 1074842182, 1471785802, 1074826703]}, 'Perm': {'d': 14, 'gens': [2003546160, 6438625326, 14377673521, 21196264608, 15335757414, 20717187409, 27430179840, 15335757360, 288]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^7:C_6', 'transitive_degree': 24, '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': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [4, 4, 4, 12, 4, 12], 'aut_gens': [[194598041208, 2878923376655, 1309154923729], [7872035678723, 9360729498643, 1309154923729], [4383866201185, 3171549825043, 12940669009], [1327879901688, 7069193101818, 19280182609], [3941824284803, 5754253859830, 4371412152529], [1340732757625, 7081599611437, 19280182609], [7964486986488, 5581406517037, 4371412152529]], 'aut_group': None, 'aut_hash': 8103672493863816722, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 18432, 'aut_permdeg': 72, 'aut_perms': [57274809251007934509142169297040852855661271136882922905506372116241856690953098889232687897759763715948, 60338182053119402293623919724497817743847569221283900226810729865806646683110270691508430337581209785402, 38461308798321027427957011875770882292266612379960776010243292861805276706919599332009586964051727732767, 21610488004326440497364916768466113866113259628490905254526212696031798146166106109526530068307594156460, 28246322834814581327676330564081672956460746710691939703600484492926818756884626063977560945491054141906, 11487939374359779003098185892017022799431868422320178528229227800558927825251340941614752252732222170933], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 6, 1, 2], [2, 6, 2, 1], [2, 9, 1, 2], [2, 12, 1, 2], [2, 18, 1, 3], [2, 48, 1, 1], [3, 8, 2, 1], [3, 16, 2, 1], [3, 32, 1, 1], [4, 4, 2, 1], [4, 12, 4, 1], [4, 36, 2, 1], [4, 48, 1, 1], [4, 48, 2, 1], [4, 144, 1, 2], [4, 144, 2, 1], [6, 8, 2, 1], [6, 8, 4, 1], [6, 16, 2, 3], [6, 24, 2, 2], [6, 24, 4, 1], [6, 32, 1, 1], [6, 32, 2, 4], [6, 48, 2, 2], [6, 64, 1, 2], [6, 192, 2, 1], [12, 16, 8, 1], [12, 48, 8, 1], [12, 64, 4, 2], [12, 192, 2, 1], [12, 192, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'A_4^2.C_2^6.C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '4608.mp', 'autcentquo_hash': 3801550586087620078, 'autcentquo_nilpotent': False, 'autcentquo_order': 4608, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^5:D_6^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 6, 4], [2, 9, 2], [2, 12, 2], [2, 18, 3], [2, 48, 1], [3, 8, 2], [3, 16, 2], [3, 32, 1], [4, 4, 2], [4, 12, 4], [4, 36, 2], [4, 48, 3], [4, 144, 4], [6, 8, 6], [6, 16, 6], [6, 24, 8], [6, 32, 9], [6, 48, 4], [6, 64, 2], [6, 192, 2], [12, 16, 8], [12, 48, 8], [12, 64, 8], [12, 192, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': None, 'commutator_count': 1, 'commutator_label': '192.1540', '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', '2.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 67, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 6, 1, 4], [2, 9, 1, 2], [2, 12, 1, 2], [2, 18, 1, 3], [2, 48, 1, 1], [3, 8, 2, 1], [3, 16, 2, 1], [3, 32, 1, 1], [4, 4, 2, 1], [4, 12, 2, 2], [4, 36, 2, 1], [4, 48, 1, 1], [4, 48, 2, 1], [4, 144, 1, 2], [4, 144, 2, 1], [6, 8, 2, 3], [6, 16, 2, 3], [6, 24, 2, 4], [6, 32, 1, 1], [6, 32, 2, 4], [6, 48, 2, 2], [6, 64, 1, 2], [6, 192, 2, 1], [12, 16, 4, 2], [12, 48, 4, 2], [12, 64, 4, 2], [12, 192, 2, 1], [12, 192, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 240, 'exponent': 12, 'exponents_of_order': [9, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 4], [12, 1, 2], [36, 1, 1]], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '576.8656', 'hash': 3093423221651684716, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [12, 12, 2], 'inner_gens': [[194598041208, 6739632748717, 4371412152529], [4371491981117, 2878923376655, 12940669009], [7958252345208, 5412283005455, 1309154923729]], 'inner_hash': 136636571613322958, 'inner_nilpotent': False, 'inner_order': 2304, 'inner_split': True, 'inner_tex': 'C_2^4.C_6^2.C_2^2', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 24], [2, 30], [4, 9], [6, 24], [9, 8], [12, 6], [18, 2], [36, 1]], 'label': '4608.co', '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': None, 'name': 'C2^2wrS3:C12', 'ngens': 3, '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 51, 'number_characteristic_subgroups': 45, 'number_conjugacy_classes': 104, 'number_divisions': 57, 'number_normal_subgroups': 45, 'number_subgroup_autclasses': 1703, 'number_subgroup_classes': 2494, 'number_subgroups': 61206, 'old_label': None, 'order': 4608, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 175], [3, 80], [4, 848], [6, 1328], [12, 2176]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[194598050448, 2878923386135, 1309154923729], [1309198476408, 3139863127055, 19280182609], [4371491993208, 3139895423375, 31697936209]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 12], [6, 4], [8, 6], [9, 4], [12, 8], [18, 4], [24, 4], [36, 1]], 'representations': {'PC': {'code': '5013127719691357252545161940365611221631732204014854950906345965399954978057925279006845650070473000707466555465661402933506866173605118368845616950982349647785092534098160867600', 'gens': [1, 4, 6, 8, 9, 10, 11], 'pres': [11, -2, -2, -3, -2, -2, 2, -3, -2, 2, 2, 2, 22, 56, 49107, 64166, 19165, 124, 238924, 103635, 20156, 17429, 20608, 29331, 14030, 973, 192, 192198, 70241, 24052, 17903, 8674, 253447, 139410, 63389, 19048, 4286, 2449, 342152, 57043, 42798, 35681, 2439, 1856, 95049, 95060, 15871, 31722, 4024, 735, 243946, 87141, 113288, 6599, 439]}, 'Perm': {'d': 16, 'gens': [194598041208, 2878923376655, 1309154923729]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\wr S_3:C_{12}', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
