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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '38880.bf', 'ambient_counter': 32, 'ambient_order': 38880, 'ambient_tex': 'C_3^2:S_3\\times S_6', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 3, 'counter': 372, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '38880.bf.810.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '810.a1', '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': 810, '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': '48.44', 'subgroup_hash': 44, 'subgroup_order': 48, 'subgroup_tex': 'C_2^2\\times C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '38880.bf', 'aut_centralizer_order': None, 'aut_label': '810.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '810.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['270.e1', '405.a1'], 'contains': ['1620.a2', '1620.c1', '1620.d1', '1620.r1', '1620.s1', '1620.bg1', '1620.bh1', '2430.a1'], 'core': '12960.a1', 'coset_action_label': None, 'count': 405, 'diagramx': [934, -1, 1499, -1], 'generators': [274467962307, 99632413684, 80884, 3674614, 693], 'label': '38880.bf.810.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '405.a1', 'old_label': '810.a1', 'projective_image': '12960.cn', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '810.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': '48.44', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [3, 2, 4], [1, 2, 20], [2, 3, 4], [1, 2, 6], [1, 2, 7], [1, 26, 29], [25, 2, 30], [1, 2, 28]], 'aut_group': '384.20089', 'aut_hash': 20089, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 10, 'aut_perms': [85945, 848929, 166680, 1961646, 1270440, 1538646, 2682390, 848928], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [3, 1, 2, 1], [4, 1, 8, 1], [6, 1, 2, 1], [6, 1, 12, 1], [12, 1, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '384.20089', 'autcent_hash': 20089, 'autcent_nilpotent': False, 'autcent_order': 384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4:S_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, 7], [3, 1, 2], [4, 1, 8], [6, 1, 14], [12, 1, 16]], 'center_label': '48.44', 'center_order': 48, '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', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 44, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 1], [4, 1, 2, 4], [6, 1, 2, 7], [12, 1, 4, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 91, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '24.15', 'hash': 44, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 48]], 'label': '48.44', 'linC_count': 5824, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 96, 'linQ_dim': 5, 'linQ_dim_count': 96, 'linR_count': 96, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C12', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 48, 'number_divisions': 24, 'number_normal_subgroups': 54, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 54, 'number_subgroups': 54, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 8], [6, 14], [12, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[3, 2, 4], [1, 2, 20], [2, 3, 4], [1, 2, 6], [1, 2, 7], [1, 26, 29], [25, 2, 30], [1, 2, 28]], 'outer_group': '384.20089', 'outer_hash': 20089, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 10, 'outer_perms': [85945, 848929, 166680, 1961646, 1270440, 1538646, 2682390, 848928], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 4]], 'representations': {'PC': {'code': 26739073, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -3, 42, 58]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [705946296092, 141602720900, 706461794101]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [10319436418, 9789111396, 6526074264, 4548416880, 6762482606]}, 'Perm': {'d': 11, 'gens': [2400, 3628800, 40320, 4, 744]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 12], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{12}', 'transitive_degree': 48, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [24, 10, 12, 4], 'aut_gens': [[80784, 243, 321844, 3674614, 138240, 93924276214], [148, 80643, 276723, 186814529973, 693, 443076757173], [45600, 148, 276628, 274471626934, 219024, 13455671734], [184054, 219474, 321987, 83784837, 322294, 94407183237], [219474, 138933, 276480, 88699035478, 276930, 175358408278]], 'aut_group': None, 'aut_hash': 940185586343533795, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 622080, 'aut_permdeg': 39, 'aut_perms': [6451987112392995459389260414160347013927683717, 3617421984017891406911333563175294466830992, 164077190581980744496036517063503329764498416, 11433359915061552333477978266584724334382906813], 'aut_phi_ratio': 60.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 15, 2, 1], [2, 45, 1, 1], [2, 135, 2, 1], [2, 405, 1, 1], [3, 1, 2, 1], [3, 6, 4, 1], [3, 40, 2, 1], [3, 40, 4, 1], [3, 240, 8, 1], [4, 90, 1, 2], [4, 810, 1, 2], [5, 144, 1, 1], [6, 9, 2, 1], [6, 15, 4, 1], [6, 45, 2, 1], [6, 90, 8, 1], [6, 120, 2, 1], [6, 120, 4, 1], [6, 135, 4, 1], [6, 270, 4, 1], [6, 360, 2, 1], [6, 360, 4, 1], [6, 405, 2, 1], [6, 720, 8, 1], [6, 1080, 2, 1], [6, 1080, 4, 1], [10, 1296, 1, 1], [12, 90, 2, 2], [12, 540, 4, 2], [12, 810, 2, 2], [15, 144, 2, 1], [15, 864, 4, 1], [30, 1296, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '\\AGL(2,3).A_6.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': None, 'autcentquo_hash': 940185586343533795, 'autcentquo_nilpotent': False, 'autcentquo_order': 622080, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\AGL(2,3).A_6.C_2^2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 15, 2], [2, 45, 1], [2, 135, 2], [2, 405, 1], [3, 1, 2], [3, 6, 4], [3, 40, 6], [3, 240, 8], [4, 90, 2], [4, 810, 2], [5, 144, 1], [6, 9, 2], [6, 15, 4], [6, 45, 2], [6, 90, 8], [6, 120, 6], [6, 135, 4], [6, 270, 4], [6, 360, 6], [6, 405, 2], [6, 720, 8], [6, 1080, 6], [10, 1296, 1], [12, 90, 4], [12, 540, 8], [12, 810, 4], [15, 144, 2], [15, 864, 4], [30, 1296, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '12960.cn', 'commutator_count': 1, 'commutator_label': '9720.bc', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '360.118'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 32, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['54.8', 1], ['720.763', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 15, 1, 2], [2, 45, 1, 1], [2, 135, 1, 2], [2, 405, 1, 1], [3, 1, 2, 1], [3, 6, 1, 4], [3, 40, 1, 2], [3, 40, 2, 2], [3, 240, 1, 8], [4, 90, 1, 2], [4, 810, 1, 2], [5, 144, 1, 1], [6, 9, 2, 1], [6, 15, 2, 2], [6, 45, 2, 1], [6, 90, 1, 8], [6, 120, 1, 2], [6, 120, 2, 2], [6, 135, 2, 2], [6, 270, 1, 4], [6, 360, 1, 2], [6, 360, 2, 2], [6, 405, 2, 1], [6, 720, 1, 8], [6, 1080, 1, 2], [6, 1080, 2, 2], [10, 1296, 1, 1], [12, 90, 2, 2], [12, 540, 1, 8], [12, 810, 2, 2], [15, 144, 2, 1], [15, 864, 1, 4], [30, 1296, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 34169688, 'exponent': 60, 'exponents_of_order': [5, 5, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[15, 0, 16], [27, 0, 8], [30, 0, 8], [48, 0, 4]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '12960.cn', 'hash': 6438938557571720405, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [3, 3, 3, 2, 2, 10], 'inner_gens': [[80784, 243, 357264, 3674850, 219024, 93924276450], [80784, 243, 357264, 3674850, 219024, 93924276450], [45364, 80784, 321844, 3847683, 322294, 93924449283], [80643, 148, 219333, 3674614, 357573, 93967821814], [45600, 80643, 138838, 3905878, 138240, 93924507478], [80643, 148, 219333, 442641069814, 357573, 93924276214]], 'inner_hash': 1596196825562955252, 'inner_nilpotent': False, 'inner_order': 12960, 'inner_split': False, 'inner_tex': 'C_3:S_3\\times S_6', 'inner_used': [1, 3, 4, 5, 6], 'irrC_degree': 15, 'irrQ_degree': 30, 'irrQ_dim': 30, 'irrR_degree': 30, 'irrep_stats': [[1, 4], [2, 8], [3, 8], [5, 8], [9, 4], [10, 20], [15, 16], [16, 2], [18, 8], [20, 8], [27, 8], [30, 8], [32, 4], [48, 4]], 'label': '38880.bf', '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': False, 'name': 'C3^2:S3*S6', 'ngens': 6, '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, 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, 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': 40, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 110, 'number_divisions': 88, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 699, 'number_subgroup_classes': 2672, 'number_subgroups': 350735, 'old_label': None, 'order': 38880, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 759], [3, 2186], [4, 1800], [5, 144], [6, 18438], [10, 1296], [12, 7920], [15, 3744], [30, 2592]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 101149216707, 45814, 45814, 45814], 'outer_gens': [[45364, 243, 219474, 3710034, 276930, 93924311634], [80784, 243, 321844, 375138132214, 138240, 274072423414], [80643, 148, 138690, 3674614, 276930, 93924276214], [219474, 138933, 357364, 3674614, 598, 93924276214], [138838, 184290, 45364, 3674614, 322294, 93924276214]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [396, 341, 186, 339, 107], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [5, 8], [6, 4], [9, 4], [10, 20], [16, 2], [18, 8], [20, 8], [30, 8], [32, 4], [54, 4], [60, 4], [96, 2]], 'representations': {'Perm': {'d': 15, 'gens': [80784, 243, 321844, 3674614, 138240, 93924276214]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:S_3\\times S_6', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
