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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '23328.gf', 'ambient_counter': 162, 'ambient_order': 23328, 'ambient_tex': 'C_3^5:(C_2^2\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 36, 'counter': 311, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '23328.gf.27.h1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '27.h1', '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': 27, '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': '864.4406', 'subgroup_hash': 4406, 'subgroup_order': 864, 'subgroup_tex': 'C_2^2:S_3^3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '23328.gf', 'aut_centralizer_order': None, 'aut_label': '27.h1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['9.d1', '9.g1'], 'contains': ['54.i1', '54.n1', '54.u1', '54.w1', '54.bw1', '54.bz1', '54.ci1', '54.cm1', '54.do1', '54.dx1', '54.dz1', '54.fi1', '54.fn1', '54.go1', '54.hc1'], 'core': '648.B', 'coset_action_label': None, 'count': 27, 'diagramx': [7298, -1, 6658, -1], 'generators': [349506, 329911, 737007977329752556, 614672336678670305, 155828506069716480000, 136, 1520741099, 482904299], 'label': '23328.gf.27.h1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.h1', 'old_label': '27.h1', 'projective_image': '23328.gf', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '27.h1', 'subgroup_fusion': None, 'weyl_group': '432.759'}
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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': 12, 'aut_gen_orders': [2, 4, 6, 6, 12, 4, 12], 'aut_gens': [[5040, 482630536, 1037836943, 39916800, 1520467200, 325, 435, 45360], [40320, 482630536, 482630543, 1480550400, 1520467200, 325, 435, 80640], [85680, 482631093, 1037836936, 1480550400, 1520467200, 348, 325, 80640], [1520552880, 482631059, 1037836943, 39916800, 1520467200, 458, 435, 80640], [40320, 482631059, 482631093, 1480550400, 1520467200, 458, 435, 45360], [1520507520, 1037837338, 1037836936, 1480550400, 1520467200, 348, 325, 45360], [5040, 482631093, 482630972, 1480550400, 1520467200, 348, 325, 80640], [85680, 482631093, 1037836936, 39916800, 1520467200, 435, 458, 45360]], 'aut_group': None, 'aut_hash': 9142008714284746175, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6912, 'aut_permdeg': 20, 'aut_perms': [93361750047965288, 1972078105163194271, 1637307595573357874, 2414222544251750133, 2063157015484634679, 140978852905047547, 241571824193475265], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 2, 1], [2, 6, 1, 1], [2, 6, 2, 1], [2, 9, 2, 1], [2, 18, 1, 1], [2, 18, 2, 1], [2, 27, 2, 1], [2, 54, 1, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [4, 6, 2, 1], [4, 18, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 1, 2], [6, 4, 2, 3], [6, 6, 4, 1], [6, 8, 1, 1], [6, 8, 2, 2], [6, 12, 2, 5], [6, 18, 2, 1], [6, 24, 2, 1], [6, 36, 1, 1], [6, 36, 2, 1], [12, 12, 2, 2], [12, 24, 2, 1], [12, 36, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3.C_2^6.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '432.741', 'autcentquo_hash': 741, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^3:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 3, 2], [2, 6, 3], [2, 9, 2], [2, 18, 3], [2, 27, 2], [2, 54, 1], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 6, 2], [4, 18, 2], [6, 2, 3], [6, 4, 8], [6, 6, 4], [6, 8, 5], [6, 12, 10], [6, 18, 2], [6, 24, 2], [6, 36, 3], [12, 12, 4], [12, 24, 2], [12, 36, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '432.759', 'commutator_count': 1, 'commutator_label': '54.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4406, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['144.154', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 6, 1, 3], [2, 9, 1, 2], [2, 18, 1, 3], [2, 27, 1, 2], [2, 54, 1, 1], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 6, 1, 2], [4, 18, 1, 2], [6, 2, 1, 3], [6, 4, 1, 8], [6, 6, 1, 4], [6, 8, 1, 5], [6, 12, 1, 10], [6, 18, 1, 2], [6, 24, 1, 2], [6, 36, 1, 3], [12, 12, 1, 4], [12, 24, 1, 2], [12, 36, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 22145760, 'exponent': 12, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '432.759', 'hash': 4406, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 2, 2, 1, 3, 3, 3], 'inner_gens': [[5040, 482630536, 1037836943, 39916800, 1520467200, 325, 435, 80640], [5040, 482630536, 1037836943, 1480550400, 1520467200, 325, 348, 45360], [5040, 482630536, 1037836943, 1480550400, 1520467200, 458, 435, 45360], [5040, 1037836936, 482630543, 39916800, 1520467200, 325, 435, 45360], [5040, 482630536, 1037836943, 39916800, 1520467200, 325, 435, 45360], [5040, 482630536, 1037837493, 39916800, 1520467200, 325, 435, 45360], [5040, 482631059, 1037836943, 39916800, 1520467200, 325, 435, 45360], [85680, 482630536, 1037836943, 39916800, 1520467200, 325, 435, 45360]], 'inner_hash': 759, 'inner_nilpotent': False, 'inner_order': 432, 'inner_split': False, 'inner_tex': 'C_2\\times S_3^3', 'inner_used': [1, 2, 3, 4, 6, 7, 8], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 28], [4, 22], [8, 6]], 'label': '864.4406', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'C2^2:S3^3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 42, 'number_characteristic_subgroups': 46, 'number_conjugacy_classes': 72, 'number_divisions': 72, 'number_normal_subgroups': 158, 'number_subgroup_autclasses': 600, 'number_subgroup_classes': 1218, 'number_subgroups': 10256, 'old_label': None, 'order': 864, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 207], [3, 26], [4, 48], [6, 414], [12, 168]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 1520467207], 'outer_gens': [[5040, 482630536, 1037836943, 39916800, 1520467200, 325, 348, 45360], [5040, 482630536, 482630543, 39916800, 1520467200, 325, 435, 80640], [1520472240, 482630536, 1037836943, 39916800, 1520467200, 458, 435, 45360], [5040, 482630543, 1037836936, 39916800, 1520467200, 435, 458, 80640]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 28], [4, 22], [8, 6]], 'representations': {'PC': {'code': 14203866604722635857611706835798508983317, 'gens': [1, 2, 3, 5, 7], 'pres': [8, -2, -2, -2, -3, -2, -3, -2, -3, 250, 66, 267, 18244, 5540, 116, 1173, 40326, 166, 36871]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729074820244, 24992899258178932, 125124615946071274, 91793122737612576]}, 'Perm': {'d': 13, 'gens': [5040, 482630536, 1037836943, 39916800, 1520467200, 325, 435, 45360]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2:S_3^3', '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': '8.5', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [6, 6, 12, 6], 'aut_gens': [[56322415181871424880, 109872558001407854978, 56444749340408370772], [209255601961526236853, 323966555037359217183, 117035811105660595000], [319230242254717686111, 323723956595763373052, 272998658484747925993], [273247662002533095697, 421040078769842167174, 319351113211425286795], [423838564771436836443, 421282615749704836383, 319230242254717632116]], 'aut_group': None, 'aut_hash': 7111735032972534564, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 69984, 'aut_permdeg': 90, 'aut_perms': [560325420625779715207351732751197598832470518688719267135120697995796285660114899072506949397829589864361133233447537909888284662488171758, 1184556615668432002274064098394001483918032466170247804454716938402001990768018493614268755731077138311974389999728402523817749586921070142, 692279061603564703806921467373666490933751579145100730250279062509682027058537025301176349657918371505637121220072766484565704161077288484, 270354858724106555253194995398892848039823307888165055439113622619410262915265136841936668765604763743959536416381572265982136545266622611], 'aut_phi_ratio': 9.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [2, 54, 1, 2], [2, 81, 1, 1], [2, 162, 1, 2], [2, 243, 1, 1], [2, 729, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 2], [3, 6, 3, 1], [3, 12, 1, 3], [3, 12, 3, 2], [3, 24, 1, 1], [3, 24, 3, 1], [3, 216, 1, 1], [3, 432, 1, 1], [4, 54, 1, 2], [4, 162, 1, 2], [6, 6, 1, 4], [6, 6, 3, 1], [6, 12, 1, 6], [6, 12, 3, 3], [6, 18, 1, 1], [6, 24, 1, 6], [6, 24, 3, 5], [6, 54, 1, 4], [6, 54, 3, 2], [6, 108, 1, 5], [6, 108, 3, 3], [6, 162, 1, 3], [6, 216, 1, 2], [6, 216, 3, 2], [6, 324, 1, 5], [6, 648, 1, 4], [6, 1296, 1, 1], [6, 1944, 1, 1], [9, 432, 1, 1], [9, 864, 1, 1], [12, 108, 1, 4], [12, 108, 3, 2], [12, 216, 1, 2], [12, 216, 3, 2], [12, 324, 1, 4], [12, 648, 1, 2], [18, 1296, 1, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_3\\times C_6^2).C_3^4.C_2^3', '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': 36, 'autcentquo_group': None, 'autcentquo_hash': 7111735032972534564, 'autcentquo_nilpotent': False, 'autcentquo_order': 69984, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(C_3\\times C_6^2).C_3^4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 2], [2, 54, 2], [2, 81, 1], [2, 162, 2], [2, 243, 1], [2, 729, 1], [3, 2, 2], [3, 4, 1], [3, 6, 5], [3, 12, 9], [3, 24, 4], [3, 216, 1], [3, 432, 1], [4, 54, 2], [4, 162, 2], [6, 6, 7], [6, 12, 15], [6, 18, 1], [6, 24, 21], [6, 54, 10], [6, 108, 14], [6, 162, 3], [6, 216, 8], [6, 324, 5], [6, 648, 4], [6, 1296, 1], [6, 1944, 1], [9, 432, 1], [9, 864, 1], [12, 108, 10], [12, 216, 8], [12, 324, 4], [12, 648, 2], [18, 1296, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '23328.gf', 'commutator_count': 1, 'commutator_label': None, 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 162, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [2, 54, 1, 2], [2, 81, 1, 1], [2, 162, 1, 2], [2, 243, 1, 1], [2, 729, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 5], [3, 12, 1, 9], [3, 24, 1, 4], [3, 216, 1, 1], [3, 432, 1, 1], [4, 54, 1, 2], [4, 162, 1, 2], [6, 6, 1, 7], [6, 12, 1, 15], [6, 18, 1, 1], [6, 24, 1, 21], [6, 54, 1, 10], [6, 108, 1, 14], [6, 162, 1, 3], [6, 216, 1, 8], [6, 324, 1, 5], [6, 648, 1, 4], [6, 1296, 1, 1], [6, 1944, 1, 1], [9, 432, 1, 1], [9, 864, 1, 1], [12, 108, 1, 10], [12, 216, 1, 8], [12, 324, 1, 4], [12, 648, 1, 2], [18, 1296, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 36, 'exponents_of_order': [6, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 6], [24, 1, 12]], 'familial': False, 'frattini_label': '27.5', 'frattini_quotient': '864.4673', 'hash': 6558491815484891589, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 6, 6], 'inner_gens': [[56322415181871424880, 420917698756438410783, 56687391021917787892], [423838564771436770813, 109872558001407854978, 272998658485223323101], [56687391021003338960, 267773632397436376455, 56444749340408370772]], 'inner_hash': 6558491815484891589, 'inner_nilpotent': False, 'inner_order': 23328, 'inner_split': True, 'inner_tex': 'C_3^5:(C_2^2\\times S_4)', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 12], [3, 8], [4, 6], [6, 48], [8, 1], [12, 48], [24, 25]], 'label': '23328.gf', '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': 'C3^5:(C2^2*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, 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': 108, 'number_characteristic_subgroups': 70, 'number_conjugacy_classes': 156, 'number_divisions': 156, 'number_normal_subgroups': 70, 'number_subgroup_autclasses': 7122, 'number_subgroup_classes': 12262, 'number_subgroups': 1200602, 'old_label': None, 'order': 23328, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 1551], [3, 890], [4, 432], [6, 12462], [9, 1296], [12, 5400], [18, 1296]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [0], 'outer_gens': [[56322415181871683801, 109872558001407854978, 56444749340408370772]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 6], [6, 48], [8, 1], [12, 48], [24, 25]], 'representations': {'PC': {'code': '6049873089019606199910298146745122310054379033542676896793069124869380881926362621092323255876448563007061093788833767407018184290187959163999145558930120525697819965783', 'gens': [1, 2, 4, 6, 8, 10, 11], 'pres': [11, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 39600, 445853, 56, 216482, 10716, 446163, 112214, 251221, 124, 188764, 522075, 177566, 413429, 453832, 113294, 8959, 192, 332646, 742913, 95211, 19454, 57031, 95058, 4792, 7179, 260, 85555, 110889, 190100, 9282, 60113, 1986346, 575013, 165571]}, 'Perm': {'d': 22, 'gens': [56322415181871424880, 109872558001407854978, 56444749340408370772]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 54, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^5:(C_2^2\\times S_4)', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
