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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '20736.ig', 'ambient_counter': 215, 'ambient_order': 20736, 'ambient_tex': 'C_6:D_6^2:S_4', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 432, 'counter': 170, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '20736.ig.24.da1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '24.da1', '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': 24, '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': '864.4723', 'subgroup_hash': 4723, 'subgroup_order': 864, 'subgroup_tex': 'C_6^3:C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '20736.ig', 'aut_centralizer_order': None, 'aut_label': '24.da1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '432.is1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.a1', '12.n1', '12.bj1', '12.bk1', '12.bl1'], 'contains': ['48.a1', '48.bs1', '48.bt1', '48.bu1', '48.dc1', '48.dd1', '48.de1', '48.df1', '72.fv1', '72.fw1', '72.fx1'], 'core': '48.a1', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [14519, 8052576094080, 106127159402880, 17764, 482630400, 3043, 4479, 118836305280], 'label': '20736.ig.24.da1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '12.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '24.da1', 'projective_image': '20736.ig', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.da1', 'subgroup_fusion': None, 'weyl_group': '144.186'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '96.231', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 840, 'aut_gen_orders': [56, 12, 56], 'aut_gens': [[886789, 47249, 607189, 1104635, 47215], [47647, 793441, 1353359, 886573, 1369013], [1633949, 1462793, 327247, 1197643, 264719], [46999, 529927, 1073417, 1105067, 265493]], 'aut_group': None, 'aut_hash': 9026865340521309599, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 278691840, 'aut_permdeg': 146, 'aut_perms': [46240388636039584050277794504209402360260364054946498381813053986718936156940022322523045103770626589759790521893847587358080575205616566802530969923439663311370581046141982425333497685988790074492269228804520450186273609650391229940328827059965554336272, 94660600899860672606969036802749679324094790030884442925622999190542265632494335959255134744629962331993788576051883351408363920087434984339847616619373881054288707496153496685643550797367426567619044969794954144049785534081088547590628665419536807240670, 43194390759992821207041848843727565611128759250069953685659908822586892098497344353328446665053228600211279789677786601216522135428580931380598716178477020677891725191564826040502707973394480771078667968719034671784790104685721571731932534120266144532396], 'aut_phi_ratio': 967680.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 15, 1], [2, 9, 16, 1], [3, 1, 2, 1], [3, 2, 4, 1], [3, 2, 8, 1], [6, 1, 30, 1], [6, 2, 60, 1], [6, 2, 120, 1], [6, 9, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_2^4.A_8\\times \\AGL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 840, 'autcent_group': '645120.e', 'autcent_hash': 6146100328714106058, 'autcent_nilpotent': False, 'autcent_order': 645120, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^5.A_8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 15], [2, 9, 16], [3, 1, 2], [3, 2, 12], [6, 1, 30], [6, 2, 180], [6, 9, 32]], 'center_label': '48.52', 'center_order': 48, 'central_product': True, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', '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': 4723, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['2.1', 4], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [2, 9, 1, 16], [3, 1, 2, 1], [3, 2, 1, 4], [3, 2, 2, 4], [6, 1, 2, 15], [6, 2, 1, 60], [6, 2, 2, 60], [6, 9, 2, 16]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 487630, 'exponent': 6, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '864.4723', 'hash': 4723, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 1, 3, 3], 'inner_gens': [[886789, 47249, 607189, 1104635, 47215], [886789, 47249, 607189, 529655, 46783], [886789, 47249, 607189, 1104635, 47215], [886789, 1198541, 607189, 1104635, 47215], [886789, 46817, 607189, 1104635, 47215]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [2, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 96], [2, 192]], 'label': '864.4723', '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': 'C6^3:C2^2', '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 288, 'number_divisions': 192, 'number_normal_subgroups': 1418, 'number_subgroup_autclasses': 70, 'number_subgroup_classes': 5024, 'number_subgroups': 15384, 'old_label': None, 'order': 864, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 159], [3, 26], [6, 678]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 840, 'outer_gen_orders': [42, 42, 14], 'outer_gen_pows': [46657, 46657, 46657], 'outer_gens': [[47305, 887075, 1166731, 265583, 358159], [886789, 1633051, 1073417, 1462885, 1104635], [793817, 1633051, 1353053, 358033, 1634075]], 'outer_group': None, 'outer_hash': 2648513413716212498, 'outer_nilpotent': False, 'outer_order': 15482880, 'outer_permdeg': 22, 'outer_perms': [186011436744222571702, 626414678696529959248, 275577509437581255093], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times C_2^4.A_8\\times S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2, 3], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 96], [4, 64]], 'representations': {'PC': {'code': 3667582641900288766155719463957, 'gens': [1, 2, 3, 5, 7], 'pres': [8, -2, -2, -2, -3, -2, -3, -2, -3, 66, 2412, 116, 2317, 20174, 166, 18447]}, 'GLZN': {'d': 2, 'p': 36, 'gens': [886789, 1197949, 1166425, 47305, 1632995, 793169, 47089, 47249]}, 'Perm': {'d': 17, 'gens': [21010447180825, 44461407518640, 65470896696240, 44460928517040, 21010447185840, 3991680, 147, 243]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6^3:C_2^2', 'transitive_degree': 288, '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': 4, 'aut_exponent': 72, 'aut_gen_orders': [4, 8, 4, 18, 4], 'aut_gens': [[2635034987002, 22418316715257], [1682816479226, 47184096162831], [69675217483317, 174363807377909], [9186376166893, 22405865938141], [50040945165237, 150396443753259], [1744564129123, 68105528876621]], 'aut_group': '5000.cq', 'aut_hash': 3482408370237741990, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 165888, 'aut_permdeg': 432, 'aut_perms': 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'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 4, 1, 1], [2, 6, 2, 1], [2, 27, 2, 1], [2, 54, 1, 1], [2, 54, 2, 1], [2, 108, 1, 1], [2, 108, 2, 1], [2, 216, 2, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 1152, 1, 1], [4, 12, 1, 1], [4, 108, 1, 1], [4, 216, 1, 1], [4, 216, 2, 1], [4, 216, 4, 1], [4, 432, 2, 2], [6, 6, 1, 1], [6, 12, 1, 2], [6, 12, 2, 1], [6, 24, 1, 3], [6, 24, 2, 2], [6, 32, 1, 1], [6, 48, 1, 1], [6, 48, 2, 2], [6, 54, 2, 1], [6, 108, 1, 1], [6, 108, 2, 1], [6, 216, 1, 1], [6, 216, 2, 1], [6, 432, 2, 1], [6, 1152, 1, 1], [9, 1152, 2, 1], [12, 24, 1, 1], [12, 48, 1, 2], [12, 96, 1, 2], [12, 216, 1, 1], [12, 216, 8, 1], [12, 432, 1, 1], [12, 432, 2, 1], [12, 432, 4, 1], [12, 864, 2, 1], [18, 1152, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^4:D_4', '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': 72, 'autcentquo_group': None, 'autcentquo_hash': 3482408370237741990, 'autcentquo_nilpotent': False, 'autcentquo_order': 165888, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^3.C_2^6.C_6.C_2^4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 4, 1], [2, 6, 2], [2, 27, 2], [2, 54, 3], [2, 108, 3], [2, 216, 2], [3, 6, 1], [3, 8, 1], [3, 12, 1], [3, 1152, 1], [4, 12, 1], [4, 108, 1], [4, 216, 7], [4, 432, 4], [6, 6, 1], [6, 12, 4], [6, 24, 7], [6, 32, 1], [6, 48, 5], [6, 54, 2], [6, 108, 3], [6, 216, 3], [6, 432, 2], [6, 1152, 1], [9, 1152, 2], [12, 24, 1], [12, 48, 2], [12, 96, 2], [12, 216, 9], [12, 432, 7], [12, 864, 2], [18, 1152, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '20736.ig', 'commutator_count': 1, 'commutator_label': '5184.me', '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', '3.1', '3.1', '3.1'], 'composition_length': 12, 'conjugacy_classes_known': True, 'counter': 215, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 4, 1, 1], [2, 6, 1, 2], [2, 27, 1, 2], [2, 54, 1, 3], [2, 108, 1, 3], [2, 216, 1, 2], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 1152, 1, 1], [4, 12, 1, 1], [4, 108, 1, 1], [4, 216, 1, 7], [4, 432, 1, 4], [6, 6, 1, 1], [6, 12, 1, 4], [6, 24, 1, 7], [6, 32, 1, 1], [6, 48, 1, 5], [6, 54, 1, 2], [6, 108, 1, 3], [6, 216, 1, 3], [6, 432, 1, 2], [6, 1152, 1, 1], [9, 1152, 1, 2], [12, 24, 1, 1], [12, 48, 1, 2], [12, 96, 1, 2], [12, 216, 1, 1], [12, 216, 2, 4], [12, 432, 1, 3], [12, 432, 2, 2], [12, 864, 1, 2], [18, 1152, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 234, 'exponent': 36, 'exponents_of_order': [8, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 8], [24, 1, 8], [48, 1, 4]], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '5184.bu', 'hash': 1337173745300760296, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [12, 18], 'inner_gens': [[2635034987002, 68441788022607], [169038714290704, 22418316715257]], 'inner_hash': 1337173745300760296, 'inner_nilpotent': False, 'inner_order': 20736, 'inner_split': None, 'inner_tex': 'C_6:D_6^2:S_4', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 2], [3, 12], [6, 26], [8, 6], [12, 22], [24, 12], [48, 4]], 'label': '20736.ig', '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': True, 'name': 'C6:D6^2:S4', 'ngens': 2, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 54, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 88, 'number_divisions': 82, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 3534, 'number_subgroup_classes': 6972, 'number_subgroups': 479708, 'old_label': None, 'order': 20736, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 991], [3, 1178], [4, 3360], [6, 3590], [9, 2304], [12, 7008], [18, 2304]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [167818689577652, 17764, 35363], 'outer_gens': [[50040945165237, 150396443753259], [9186376166893, 22405865938141], [69675217483317, 174363807377909]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 8, 'outer_perms': [19090, 13015, 5490], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 8, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [3, 12], [6, 18], [8, 6], [12, 22], [24, 14], [48, 4]], 'representations': {'PC': {'code': '599012462365884545933224386505738818633661518116767551769737691155657248361374229952293388591132966860794328601612988551142139957745161315631281089791011668458919323643174185377831940034879720072750068245', 'gens': [1, 2, 4, 5, 6, 7, 9, 11], 'pres': [12, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 2, 3, 156960, 228913, 61, 523874, 136538, 499395, 591, 124875, 312136, 26488, 198725, 425969, 99389, 934422, 546858, 425154, 906, 246, 110599, 55315, 331807, 835, 1347848, 375212, 290660, 6548, 3320, 320, 23049, 829461, 5793, 5829, 2961, 2280970, 812614, 95866, 23842, 394, 1990667, 27671, 82979, 20819]}, 'Perm': {'d': 17, 'gens': [2635034987002, 22418316715257]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6:D_6^2:S_4', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
