-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '256.53038', 'ambient_counter': 53038, 'ambient_order': 256, 'ambient_tex': 'C_2^3\\times C_4\\times C_8', 'central': True, 'central_factor': False, 'centralizer_order': 256, 'characteristic': False, 'core_order': 4, 'counter': 53, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '256.53038.64.g1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '64.g1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '64.260', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 260, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 64, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^4\\times C_4', 'simple': False, 'solvable': True, 'special_labels': ['C6'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '256.53038', 'aut_centralizer_order': 22020096, 'aut_label': '64.g1', 'aut_quo_index': 30, 'aut_stab_index': 2, 'aut_weyl_group': '2.1', 'aut_weyl_index': 44040192, 'centralizer': '1.a1', 'complements': [], 'conjugacy_class_count': 2, 'contained_in': ['32.b1', '32.j1', '32.k1'], 'contains': ['128.c1'], 'core': '64.g1', 'coset_action_label': None, 'count': 2, 'diagramx': [2939, 2939, 2939, 2939], 'generators': [64], 'label': '256.53038.64.g1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '64.g1', 'normal_contained_in': ['32.b1', '32.j1', '32.k1'], 'normal_contains': ['128.c1'], 'normalizer': '1.a1', 'old_label': '64.g1', 'projective_image': '64.260', 'quotient_action_image': '1.1', 'quotient_action_kernel': '64.260', 'quotient_action_kernel_order': 64, 'quotient_fusion': None, 'short_label': '64.g1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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, 1], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '256.53038', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [28, 12, 24], 'aut_gens': [[1, 2, 4, 8, 32], [132, 145, 22, 91, 105], [3, 151, 145, 11, 232], [17, 23, 130, 220, 236]], 'aut_group': None, 'aut_hash': 5302274406425492967, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 88080384, 'aut_permdeg': 128, 'aut_perms': [126103313064787726297673456756029641913976405645893512836830131611665225364159985861987983552317302532149244643009907824666744676311697700611182673569400029997406626781853373686128909913315545741060623219287584510979, 315843097489821333113988489494114294144521031474269653761040257182033755087624910429737741122596382369476657442079695082199563467738095223944868483933751874748833133234155699347429977535285886545983409360161158026118, 328331905692079261377285010693327746891749928996802904345869226609532956205629063555521669662717194430043208552919329273140780600241089275654978697351636332732996474814969449330613621759740535897333583007449330226123], 'aut_phi_ratio': 688128.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 1, 28, 1], [4, 1, 4, 1], [4, 1, 28, 1], [4, 1, 64, 1], [8, 1, 128, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6.C_2^5.C_2^6.C_2^2.\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': None, 'autcent_hash': 5302274406425492967, 'autcent_nilpotent': False, 'autcent_order': 88080384, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^6.C_2^5.C_2^6.C_2^2.\\PSL(2,7)', '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, 31], [4, 1, 96], [8, 1, 128]], 'center_label': '256.53038', 'center_order': 256, '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', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 53038, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['4.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 31], [4, 1, 2, 48], [8, 1, 4, 32]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3720, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '32.51', 'hash': 53038, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8, 32], [1, 2, 4, 8, 32], [1, 2, 4, 8, 32], [1, 2, 4, 8, 32], [1, 2, 4, 8, 32]], '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, 256]], 'label': '256.53038', 'linC_count': None, 'linC_degree': 5, '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': 'C2^3*C4*C8', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 256, 'number_divisions': 112, 'number_normal_subgroups': 2319, 'number_subgroup_autclasses': 57, 'number_subgroup_classes': 2319, 'number_subgroups': 2319, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 31], [4, 96], [8, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [4, 12, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[145, 7, 132, 156, 41], [3, 20, 22, 218, 248], [129, 19, 5, 159, 227]], 'outer_group': None, 'outer_hash': 5302274406425492967, 'outer_nilpotent': False, 'outer_order': 88080384, 'outer_permdeg': 128, 'outer_perms': [29180317846392730731093894385882836941262401930341658036602928573050217665124743362914024937616708969923314238752498748025280256969264786332907035958949965725739538066743386754338229435299646818868625844388447784189, 364433753495042476832227503163485376141778740587285301167239343630879093471953620963098526586594152430625816779517969064956146975037557603853537238583587230805040307040170791171624258992828510255429887308488135443384, 299080663679963708406565722778251330040136564392635872246263202292287305185709798342175570857748475901400060930719230985708780384987627944437384569582422716378543999547845425114728746542408766396592721463021183669461], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^6.C_2^5.C_2^6.C_2^2.\\PSL(2,7)', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 4, 8], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 48], [4, 32]], 'representations': {'PC': {'code': 2269666877636712, 'gens': [1, 2, 3, 4, 6], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 91, 141, 166]}, 'GLZN': {'d': 2, 'p': 24, 'gens': [13969, 18641, 76325, 158987, 186925, 136429, 76337, 14113]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [4105, 39041, 36873, 13359, 4161, 22669, 12291, 4225]}, 'Perm': {'d': 18, 'gens': [40176, 127008000, 6227020800, 355687428096000, 1307674368000, 16582, 40279680, 5167]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4, 8], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3\\times C_4\\times C_8', 'transitive_degree': 256, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '64.260', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 840, 'aut_gen_orders': [4, 30], 'aut_gens': [[1, 2, 4, 8, 16], [41, 12, 10, 13, 29], [34, 4, 15, 39, 20]], 'aut_group': '10321920.d', 'aut_hash': 4172769537597284401, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10321920, 'aut_permdeg': 32, 'aut_perms': [112652731098387796720859227842634419, 120074902283433332098813261435304397], 'aut_phi_ratio': 322560.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 30, 1], [4, 1, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5.C_2^4.A_8', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 840, 'autcent_group': '10321920.d', 'autcent_hash': 4172769537597284401, 'autcent_nilpotent': False, 'autcent_order': 10321920, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^5.C_2^4.A_8', '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, 31], [4, 1, 32]], 'center_label': '64.260', 'center_order': 64, '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', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 260, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 4], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 31], [4, 1, 2, 16]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 31, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '32.51', 'hash': 260, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8, 16], [1, 2, 4, 8, 16], [1, 2, 4, 8, 16], [1, 2, 4, 8, 16], [1, 2, 4, 8, 16]], '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, 64]], 'label': '64.260', 'linC_count': None, 'linC_degree': 5, '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': 'C2^4*C4', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 64, 'number_divisions': 48, 'number_normal_subgroups': 681, 'number_subgroup_autclasses': 15, 'number_subgroup_classes': 681, 'number_subgroups': 681, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 31], [4, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 840, 'outer_gen_orders': [4, 30], 'outer_gen_pows': [0, 0], 'outer_gens': [[41, 12, 10, 13, 29], [34, 4, 15, 39, 20]], 'outer_group': '10321920.d', 'outer_hash': 4172769537597284401, 'outer_nilpotent': False, 'outer_order': 10321920, 'outer_permdeg': 32, 'outer_perms': [112652731098387796720859227842634419, 120074902283433332098813261435304397], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^5.C_2^4.A_8', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 4], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 16]], 'representations': {'PC': {'code': 1048592, 'gens': [1, 2, 3, 4, 5], 'pres': [6, -2, 2, 2, 2, 2, -2, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729084386182, 125101749995526650, 125101750005091126, 25038659807093174, 24992906222183252]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [5601, 19949, 8645, 2665, 1801, 9587]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [1831, 2853, 1827, 545, 659, 1539]}, 'Perm': {'d': 12, 'gens': [22, 39916800, 362880, 5040, 120, 7]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 4], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4\\times C_4', 'transitive_degree': 64, 'wreath_data': None, 'wreath_product': False}
