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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '960.6296', 'ambient_counter': 6296, 'ambient_order': 960, 'ambient_tex': 'C_{10}.\\GL(2,\\mathbb{Z}/4)', 'central': False, 'central_factor': False, 'centralizer_order': 10, 'characteristic': False, 'core_order': 120, 'counter': 9, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.6296.4.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '4.d1.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': 4, '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': '240.102', 'subgroup_hash': 102, 'subgroup_order': 240, 'subgroup_tex': 'C_{10}.S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.6296', 'aut_centralizer_order': 4, 'aut_label': '4.d1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '192.1469', 'aut_weyl_index': 8, 'centralizer': '96.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1'], 'contains': ['8.a1.a1', '12.n1.a1', '16.c1.a1', '20.d1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [8888, -1, 8680, -1, 8146, -1, 8702, -1], 'generators': [3, 480, 8, 120, 528, 720], 'label': '960.6296.4.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.c1.a1', 'old_label': '4.d1.a1', 'projective_image': '96.195', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.d1.a1', 'subgroup_fusion': None, 'weyl_group': '48.48'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 12, 2, 2], 'aut_gens': [[93811, 493131, 804384], [175277, 431381, 804384], [860493, 493131, 804384], [233493, 493131, 863968], [175277, 236057, 804384], [93811, 193929, 804384]], 'aut_group': '192.1469', 'aut_hash': 1469, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 10, 'aut_perms': [40320, 1, 45510, 367920, 1174320], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 8, 1, 1], [4, 6, 1, 1], [4, 12, 1, 1], [5, 1, 4, 1], [6, 8, 1, 1], [8, 6, 2, 1], [10, 1, 4, 1], [15, 8, 4, 1], [20, 6, 4, 1], [20, 12, 4, 1], [30, 8, 4, 1], [40, 6, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 8, 1], [4, 6, 1], [4, 12, 1], [5, 1, 4], [6, 8, 1], [8, 6, 2], [10, 1, 4], [15, 8, 4], [20, 6, 4], [20, 12, 4], [30, 8, 4], [40, 6, 8]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '24.12', 'commutator_count': 1, 'commutator_label': '24.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 102, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['48.28', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 8, 1, 1], [4, 6, 1, 1], [4, 12, 1, 1], [5, 1, 4, 1], [6, 8, 1, 1], [8, 6, 2, 1], [10, 1, 4, 1], [15, 8, 4, 1], [20, 6, 4, 1], [20, 12, 4, 1], [30, 8, 4, 1], [40, 6, 8, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 108, 'exponent': 120, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[2, 0, 8], [4, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '120.37', 'hash': 102, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 3, 1], 'inner_gens': [[93811, 922095, 804384], [233493, 493131, 804384], [93811, 493131, 804384]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'S_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 10], [2, 15], [3, 10], [4, 5]], 'label': '240.102', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C10.S4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 40, 'number_divisions': 14, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 26, 'number_subgroups': 70, 'old_label': None, 'order': 240, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 18], [5, 4], [6, 8], [8, 12], [10, 4], [15, 32], [20, 72], [30, 32], [40, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [29792, 29792], 'outer_gens': [[860493, 493131, 804384], [93811, 493131, 863968]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [3, 2], [4, 4], [8, 1], [12, 2], [16, 2]], 'representations': {'PC': {'code': 80709308904830168126992709308675, 'gens': [1, 2, 3, 4], 'pres': [6, -2, -3, -2, 2, -2, -5, 720, 49, 3350, 548, 374, 3171, 225, 543, 69, 88]}, 'GLFp': {'d': 2, 'p': 31, 'gens': [93811, 493131, 804384]}, 'Perm': {'d': 21, 'gens': [2824580909745750960, 33, 40600003817853360, 5374589565653728560, 7952930666349143040, 10514844129330747360]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}.S_4', 'transitive_degree': 80, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [4, 2, 6, 4, 2, 2, 2], 'aut_gens': [[1, 2, 8, 24, 240], [1, 2, 8, 552, 240], [1, 2, 16, 744, 720], [1, 14, 8, 264, 600], [5, 2, 8, 24, 240], [1, 482, 608, 504, 240], [1, 242, 248, 504, 720], [1, 2, 608, 504, 240]], 'aut_group': '1536.408633772', 'aut_hash': 3864209259142932085, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 32, 'aut_perms': [56566138520, 57985020500080302669175429102807753, 255312200644786393975668904070428165, 5579163126368227592347594268851200, 230673621330612118255812696966550033, 230611468962926996033167944710541504, 123617027765288699846318063072402599], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [2, 6, 1, 1], [3, 8, 1, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 12, 1, 1], [4, 12, 2, 1], [4, 24, 1, 1], [5, 1, 4, 1], [6, 8, 1, 1], [6, 16, 2, 1], [8, 24, 1, 2], [10, 1, 4, 1], [10, 4, 4, 1], [10, 6, 4, 1], [12, 16, 1, 1], [15, 8, 4, 1], [20, 2, 4, 1], [20, 6, 4, 1], [20, 12, 4, 1], [20, 12, 8, 1], [20, 24, 4, 1], [30, 8, 4, 1], [30, 16, 8, 1], [40, 24, 4, 2], [60, 16, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times \\GL(2,\\mathbb{Z}/4):C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '96.226', 'autcentquo_hash': 226, 'autcentquo_nilpotent': False, 'autcentquo_order': 96, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^2\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 1], [2, 6, 1], [3, 8, 1], [4, 2, 1], [4, 6, 1], [4, 12, 3], [4, 24, 1], [5, 1, 4], [6, 8, 1], [6, 16, 2], [8, 24, 2], [10, 1, 4], [10, 4, 4], [10, 6, 4], [12, 16, 1], [15, 8, 4], [20, 2, 4], [20, 6, 4], [20, 12, 12], [20, 24, 4], [30, 8, 4], [30, 16, 8], [40, 24, 8], [60, 16, 4]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '96.195', 'commutator_count': 1, 'commutator_label': '48.33', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 6296, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['192.989', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [2, 6, 1, 1], [3, 8, 1, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 12, 1, 3], [4, 24, 1, 1], [5, 1, 4, 1], [6, 8, 1, 1], [6, 16, 2, 1], [8, 24, 1, 2], [10, 1, 4, 1], [10, 4, 4, 1], [10, 6, 4, 1], [12, 16, 1, 1], [15, 8, 4, 1], [20, 2, 4, 1], [20, 6, 4, 1], [20, 12, 4, 3], [20, 24, 4, 1], [30, 8, 4, 1], [30, 16, 8, 1], [40, 24, 4, 2], [60, 16, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 108, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 8], [8, 0, 4]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '240.196', 'hash': 6296, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 3, 2, 2], 'inner_gens': [[1, 6, 8, 24, 240], [485, 2, 16, 744, 720], [1, 18, 8, 384, 840], [1, 242, 848, 24, 720], [1, 482, 608, 504, 240]], 'inner_hash': 195, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': True, 'inner_tex': '\\GL(2,\\mathbb{Z}/4)', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 20], [2, 25], [3, 20], [4, 10], [6, 5], [8, 5]], 'label': '960.6296', 'linC_count': 8, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 8, 'linQ_dim': 12, 'linQ_dim_count': 8, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C10.GL(2,Z/4)', '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, 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': 30, 'number_characteristic_subgroups': 26, 'number_conjugacy_classes': 85, 'number_divisions': 32, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 132, 'number_subgroup_classes': 138, 'number_subgroups': 554, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 11], [3, 8], [4, 68], [5, 4], [6, 40], [8, 48], [10, 44], [12, 16], [15, 32], [20, 272], [30, 160], [40, 192], [60, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 2, 608, 504, 240], [1, 2, 16, 744, 720], [1, 2, 8, 552, 240]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 37, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [3, 4], [4, 7], [6, 1], [8, 4], [12, 4], [16, 3], [24, 1], [32, 1]], 'representations': {'PC': {'code': 1419985872820318798337820358041284952102537499377968, 'gens': [1, 2, 4, 5, 7], 'pres': [8, 2, 2, 2, 3, 2, 5, 2, 2, 97, 41, 11618, 2898, 267, 14892, 1948, 116, 525, 20174, 5910, 1718, 166]}, 'Perm': {'d': 37, 'gens': [414605846472394315090196669953479638243040, 54340744405032748327862850971759747036640, 33, 807247056715814191307200032658807503054720, 12375647053998074279798413852439154000, 1191302863749538458956637809207730196802400, 1573991423259863723356434312080777863340880, 1955738892294703739391539048885530182223440]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}.\\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
