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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1728.47489', 'ambient_counter': 47489, 'ambient_order': 1728, 'ambient_tex': '(Q_8\\times C_3^2):S_4', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 288, 'counter': 5, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.47489.3.c1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '3.c1.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': 3, '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': '576.8480', 'subgroup_hash': 8480, 'subgroup_order': 576, 'subgroup_tex': 'C_3\\times Q_8:S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.47489', 'aut_centralizer_order': 2, 'aut_label': '3.c1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '768.1089108', 'aut_weyl_index': 18, 'centralizer': '288.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.b1.a1', '9.a1.a1', '9.c1.a1', '12.i1.a1', '12.m1.a1', '12.m1.b1'], 'core': '6.b1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [4836, -1, 5056, -1, 6839, -1, 5689, -1], 'generators': [87091200, 458, 40367945520, 20801370240, 160796160, 7625520, 6314117040, 11335619], 'label': '1728.47489.3.c1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.c1.a1', 'old_label': '3.c1.a1', 'projective_image': '288.1036', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3.c1.a1', 'subgroup_fusion': None, 'weyl_group': '96.227'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', '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, 2, 2, 3, 2, 2, 2, 2], 'aut_gens': [[1260240, 3, 1458024, 1860720, 85944, 4578600, 8799984, 12822720], [11452464, 3, 1301760, 1860720, 1946280, 26910720, 9525600, 12822720], [1260240, 4, 1458024, 1860720, 85944, 4578600, 8799984, 12822720], [11654040, 3, 1458024, 1860720, 85944, 4578600, 8799984, 12822720], [11654040, 4, 1458024, 1860720, 85944, 21976560, 15972624, 12822720], [1988040, 3, 1623024, 1946280, 1860720, 8799984, 26349864, 12822720], [25901424, 3, 7753704, 11329920, 85944, 21936504, 8799984, 12822720], [27682104, 3, 18205920, 1860720, 12782280, 21936504, 16698240, 12822720], [364560, 3, 2388984, 1860720, 85944, 4578600, 16698240, 12822720], [364560, 3, 1623024, 1860720, 85944, 21936504, 8799984, 12822720]], 'aut_group': '768.1089108', 'aut_hash': 1089108, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 768, 'aut_permdeg': 26, 'aut_perms': [84685537159200051118525479, 355687428096000, 294709156553545243827493313, 294462946665049326174706305, 29637533914014157580655163, 168259688421391000129308221, 43448682740438859031410621, 62887847007794877065054942, 75310345805869393922721437], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 12, 1, 1], [2, 24, 1, 1], [3, 1, 2, 1], [3, 32, 1, 1], [3, 32, 2, 1], [4, 6, 2, 1], [4, 12, 2, 1], [6, 1, 2, 1], [6, 6, 2, 1], [6, 12, 2, 1], [6, 24, 2, 1], [6, 32, 1, 1], [6, 32, 2, 1], [8, 24, 2, 1], [12, 6, 4, 1], [12, 12, 4, 1], [24, 24, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5:S_4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '192.955', 'autcentquo_hash': 955, 'autcentquo_nilpotent': False, 'autcentquo_order': 192, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^3:S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 1], [2, 12, 1], [2, 24, 1], [3, 1, 2], [3, 32, 3], [4, 6, 2], [4, 12, 2], [6, 1, 2], [6, 6, 2], [6, 12, 2], [6, 24, 2], [6, 32, 3], [8, 24, 2], [12, 6, 4], [12, 12, 4], [24, 24, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '96.227', 'commutator_count': 1, 'commutator_label': '96.204', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 8480, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['192.1494', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 12, 1, 1], [2, 24, 1, 1], [3, 1, 2, 1], [3, 32, 1, 1], [3, 32, 2, 1], [4, 6, 1, 2], [4, 12, 1, 2], [6, 1, 2, 1], [6, 6, 2, 1], [6, 12, 2, 1], [6, 24, 2, 1], [6, 32, 1, 1], [6, 32, 2, 1], [8, 24, 1, 2], [12, 6, 2, 2], [12, 12, 2, 2], [24, 24, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 72, 'exponent': 24, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 4], [8, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '288.1035', 'hash': 8480, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 1, 3, 2, 2, 2, 2, 1], 'inner_gens': [[1260240, 3, 491040, 85944, 1860720, 16698240, 21936504, 12822720], [1260240, 3, 1458024, 1860720, 85944, 4578600, 8799984, 12822720], [2309304, 3, 1458024, 85944, 1946280, 19658424, 4578600, 12822720], [364560, 3, 1623024, 1860720, 85944, 21936504, 8799984, 12822720], [364560, 3, 2388984, 1860720, 85944, 4578600, 16698240, 12822720], [25901424, 3, 15086184, 11329920, 85944, 4578600, 16698240, 12822720], [20061360, 3, 25458744, 1860720, 12782280, 21936504, 8799984, 12822720], [1260240, 3, 1458024, 1860720, 85944, 4578600, 8799984, 12822720]], 'inner_hash': 227, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': True, 'inner_tex': 'C_2^2:S_4', 'inner_used': [1, 3, 4, 6], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 6], [2, 3], [3, 18], [4, 6], [6, 3], [8, 3]], 'label': '576.8480', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3*Q8:S4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 39, 'number_divisions': 26, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 126, 'number_subgroup_classes': 167, 'number_subgroups': 1068, 'old_label': None, 'order': 576, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 43], [3, 98], [4, 36], [6, 182], [8, 48], [12, 72], [24, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1260240, 3, 1458024, 1860720, 85944, 21976560, 15972624, 12822720], [1260240, 4, 1458024, 1860720, 85944, 21976560, 15972624, 12822720], [11654040, 3, 1458024, 1860720, 85944, 4578600, 8799984, 12822720]], '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': 6, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [3, 6], [4, 3], [6, 7], [8, 3], [12, 1], [16, 1]], 'representations': {'PC': {'code': 13538381408781810608177528349728246827683016191056, 'gens': [1, 2, 3, 5, 6, 8], 'pres': [8, 2, 3, 2, 3, 2, 2, 2, 2, 65, 3602, 1162, 66, 411, 9732, 11237, 2173, 1749, 325, 141, 29959, 3471, 4631]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101736460791671, 108470307958634217]}, 'Perm': {'d': 11, 'gens': [1260240, 3, 1458024, 1860720, 85944, 4578600, 8799984, 12822720]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times Q_8:S_4', '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 12, 12, 4, 6, 4], 'aut_gens': [[6391486216, 325, 51609948, 435, 7625520, 83467440, 13941935280, 20801370240, 6314117040], [211604972, 458, 46470498480, 435, 6227025840, 83467440, 46670803200, 32732869680, 6314117040], [6238351379, 458, 32667264348, 348, 6234641280, 7625520, 32772781440, 13945559040, 6314117040], [171683132, 325, 46510410240, 435, 6227025840, 83467440, 46670803200, 20801370240, 6314117040], [32655939256, 458, 25949912115, 348, 87091200, 6234641280, 46594961280, 20761458480, 6314117040], [6354392339, 458, 20600697948, 348, 6234641280, 87091200, 20761458480, 40443787440, 6314117040], [32695851452, 325, 25986200115, 348, 87091200, 6234641280, 46670803200, 32732869680, 6314117040]], 'aut_group': None, 'aut_hash': 5245665785324382618, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 13824, 'aut_permdeg': 74, 'aut_perms': [95593370222048201658197174698030093496432455729736239481069906208113321259067628591908845367405063689874373, 321153099706613643631047890708067017479137864270637641727489616148145915848731539531378746952509038583613857, 144580708142233766189053431697175243867775773744454197487030317515417056489535111081811117472042025758828664, 56912994490689234335095274005539205362818448925135551996555778562190084837540102671236000379214425651121948, 71105409729367972161169721720051873433424203819175573988257818177147958635357971488329572515252629948495414, 137698588937595558944510409000237651321476787086301569312710613796208525576812245063963103194422912067720814], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 12, 1, 1], [2, 72, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 32, 3, 1], [3, 32, 6, 1], [4, 6, 2, 1], [4, 36, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 6, 2, 1], [6, 12, 1, 1], [6, 12, 2, 3], [6, 12, 4, 1], [6, 32, 3, 1], [6, 32, 6, 1], [6, 72, 2, 1], [8, 72, 2, 1], [12, 6, 4, 1], [12, 12, 2, 1], [12, 12, 4, 1], [12, 36, 4, 1], [24, 72, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_2^4:\\He_3.C_2^4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '3456.gu', 'autcentquo_hash': 3056348276295266572, 'autcentquo_nilpotent': False, 'autcentquo_order': 3456, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.(C_2^2\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 1], [2, 12, 1], [2, 72, 1], [3, 1, 2], [3, 2, 3], [3, 32, 9], [4, 6, 2], [4, 36, 2], [6, 1, 2], [6, 2, 3], [6, 6, 2], [6, 12, 11], [6, 32, 9], [6, 72, 2], [8, 72, 2], [12, 6, 4], [12, 12, 6], [12, 36, 4], [24, 72, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '288.1036', 'commutator_count': 2, 'commutator_label': '288.987', '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', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 47489, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['576.8487', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 12, 1, 1], [2, 72, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 32, 1, 3], [3, 32, 2, 3], [4, 6, 1, 2], [4, 36, 1, 2], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 6, 2, 1], [6, 12, 1, 1], [6, 12, 2, 5], [6, 32, 1, 3], [6, 32, 2, 3], [6, 72, 2, 1], [8, 72, 1, 2], [12, 6, 2, 2], [12, 12, 1, 2], [12, 12, 2, 2], [12, 36, 2, 2], [24, 72, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 152880, 'exponent': 24, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 0, 6]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4709', 'hash': 47489, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 1, 3, 3, 2, 2, 2, 2, 1], 'inner_gens': [[6391486216, 325, 160796595, 348, 87091200, 83467440, 46670803200, 32732869680, 6314117040], [6391486216, 325, 51609948, 435, 7625520, 83467440, 13941935280, 20801370240, 6314117040], [6267301052, 325, 51609948, 435, 83467440, 87091200, 20801370240, 40367945520, 6314117040], [6391486739, 325, 51609948, 435, 7625520, 83467440, 13941935280, 20801370240, 6314117040], [6431408056, 325, 131085708, 435, 7625520, 83467440, 13941935280, 32732869680, 6314117040], [6391486216, 325, 134709468, 435, 7625520, 83467440, 25880682240, 20801370240, 6314117040], [20717182216, 325, 46543719948, 435, 7625520, 6234641280, 13941935280, 32732869680, 6314117040], [211604536, 325, 26074097628, 435, 6310483200, 83467440, 25880682240, 20801370240, 6314117040], [6391486216, 325, 51609948, 435, 7625520, 83467440, 13941935280, 20801370240, 6314117040]], 'inner_hash': 1036, 'inner_nilpotent': False, 'inner_order': 288, 'inner_split': True, 'inner_tex': '(C_2\\times C_6):S_4', 'inner_used': [1, 3, 4, 5, 7], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 6], [2, 12], [3, 18], [4, 6], [6, 18], [8, 12]], 'label': '1728.47489', 'linC_count': 48, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 18, 'linQ_dim': 8, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': '(Q8*C3^2):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, 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': 20, 'number_conjugacy_classes': 72, 'number_divisions': 47, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': 282, 'number_subgroup_classes': 496, 'number_subgroups': 5270, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 91], [3, 296], [4, 84], [6, 584], [8, 144], [12, 240], [24, 288]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 3], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[40285292, 325, 207930675, 435, 7625520, 87091200, 25880682240, 40367945520, 6314117040], [40285292, 458, 207930675, 348, 7625520, 87091200, 25880682240, 40367945520, 6314117040], [211604536, 458, 51609948, 435, 7625520, 83467440, 25877058480, 32772781440, 6314117040], [40285292, 325, 207930675, 348, 7625520, 87091200, 13945559040, 46594961280, 6314117040], [6391486739, 325, 51609600, 435, 7625520, 83467440, 13941935280, 20801370240, 6314117040]], 'outer_group': '48.51', 'outer_hash': 51, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 9, 'outer_perms': [41041, 90720, 24, 131784, 3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_6', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [3, 6], [4, 6], [6, 10], [8, 6], [12, 7], [16, 4]], 'representations': {'PC': {'code': 197764571922485565482195931318246868713477144030777891374282791073, 'gens': [1, 2, 3, 5, 6, 8], 'pres': [9, 2, 3, 2, 3, 2, 2, 2, 2, 3, 73, 4700, 1307, 74, 6051, 462, 19453, 4865, 6332, 1967, 365, 158, 23335, 11680, 5209, 214]}, 'Perm': {'d': 14, 'gens': [6391486216, 325, 51609948, 435, 7625520, 83467440, 13941935280, 20801370240, 6314117040]}}, 'schur_multiplier': [2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(Q_8\\times C_3^2):S_4', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
