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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1728.46260', 'ambient_counter': 46260, 'ambient_order': 1728, 'ambient_tex': '\\SL(2,3).\\SOPlus(4,2)', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 2, 'counter': 123, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.46260.72.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '72.b1.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': 72, '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': '24.9', 'subgroup_hash': 9, 'subgroup_order': 24, 'subgroup_tex': 'C_2\\times C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.46260', 'aut_centralizer_order': 48, 'aut_label': '72.b1', 'aut_quo_index': None, 'aut_stab_index': 36, 'aut_weyl_group': '8.5', 'aut_weyl_index': 1728, 'centralizer': '72.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.b1.a1', '18.a1.a1', '36.a1.a1'], 'contains': ['144.a1.a1', '144.i1.a1', '144.i1.b1', '216.c1.a1'], 'core': '864.a1.a1', 'coset_action_label': None, 'count': 36, 'diagramx': [1708, -1, 243, -1, 9280, -1, 362, -1], 'generators': [6, 864, 8, 12], 'label': '1728.46260.72.b1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '36.a1.a1', 'old_label': '72.b1.a1', 'projective_image': '864.4669', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '72.b1.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': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2], 'aut_gens': [[1, 2], [13, 2], [1, 23], [1, 10], [1, 14]], 'aut_group': '16.11', 'aut_hash': 11, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 6, 'aut_perms': [288, 6, 127, 126], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 2, 1], [4, 1, 4, 1], [6, 1, 2, 1], [6, 1, 4, 1], [12, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.11', 'autcent_hash': 11, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times D_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, 3], [3, 1, 2], [4, 1, 4], [6, 1, 6], [12, 1, 8]], 'center_label': '24.9', 'center_order': 24, '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', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [4, 1, 2, 2], [6, 1, 2, 3], [12, 1, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '12.5', 'hash': 9, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 24]], 'label': '24.9', 'linC_count': 96, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 4, 'linQ_dim': 4, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C12', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 24, 'number_divisions': 12, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [12, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[13, 2], [1, 23], [1, 10], [1, 14]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [288, 6, 127, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 2]], 'representations': {'PC': {'code': 221281, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -3, 21, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931072, 26129103]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [10993, 4396]}, 'Perm': {'d': 9, 'gens': [2400, 40320, 4, 744]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{12}', '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [12, 8, 6, 6, 4], 'aut_gens': [[1, 2, 24, 144], [1317, 1586, 816, 840], [19, 1330, 1224, 1128], [9, 866, 1344, 792], [37, 1142, 1560, 144], [117, 394, 816, 1272]], 'aut_group': None, 'aut_hash': 7536537919236613067, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 13824, 'aut_permdeg': 144, 'aut_perms': [4746126163832968100969164475640017732536798199509190835108390129057979281912673250975155433603238933140244945265669062517161600995152776366816114914391661800737594719869852855521140156518131215171193019168533206738508575638032761281952540238817800047, 2604569247523907598450079365116549795159417301799665139353912609527168268439906824222835454338238379456124733762578908275783061955228718862471894969976676650825278293309337469895354258128506969758947683254487196354760949572482141511354516951404672578, 4332064579387831671162490183906427290102943278632884220651807096760378995101847933964201273873954064935581845126527886817220814106510179994603686524657228320157473444666851120272318675816045907478041939188611775289299725423803390590717748657198228323, 5426390519701573510667341170625173952351308470991588646904172977034895536226520880752275579286464639873905618840277200570149193516333133733040689446469199489915503284969722172083652778596929401566241449616955934061881091032194456294496789158260681064, 4030431821253144008808581278147282888170774162706807223343458708738553204874617867815434430133363873774489487951223402982401297548199196060296095277989685231239715489762017158778299042650869116130703677958254917565387940944731140921845593426194229421], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 4, 2, 1], [3, 8, 1, 1], [3, 32, 2, 1], [4, 6, 1, 1], [4, 18, 2, 1], [4, 54, 1, 1], [4, 72, 2, 1], [4, 108, 1, 1], [6, 4, 2, 1], [6, 8, 1, 1], [6, 32, 2, 1], [6, 72, 1, 2], [8, 36, 4, 1], [12, 24, 2, 1], [12, 72, 4, 1], [12, 144, 2, 1], [24, 72, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.(C_6\\times D_4).C_2^3', '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': 24, 'autcentquo_group': '3456.jg', 'autcentquo_hash': 5305840276193666630, 'autcentquo_nilpotent': False, 'autcentquo_order': 3456, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9:C_2\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 4, 2], [3, 8, 1], [3, 32, 2], [4, 6, 1], [4, 18, 2], [4, 54, 1], [4, 72, 2], [4, 108, 1], [6, 4, 2], [6, 8, 1], [6, 32, 2], [6, 72, 2], [8, 36, 4], [12, 24, 2], [12, 72, 4], [12, 144, 2], [24, 72, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '864.4669', 'commutator_count': 1, 'commutator_label': '432.626', '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': 46260, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 4, 1, 2], [3, 8, 1, 1], [3, 32, 1, 2], [4, 6, 1, 1], [4, 18, 1, 2], [4, 54, 1, 1], [4, 72, 1, 2], [4, 108, 1, 1], [6, 4, 1, 2], [6, 8, 1, 1], [6, 32, 1, 2], [6, 72, 1, 2], [8, 36, 2, 2], [12, 24, 1, 2], [12, 72, 2, 2], [12, 144, 1, 2], [24, 72, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 24, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, -1, 4], [16, -1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4669', 'hash': 46260, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 12, 6, 6], 'inner_gens': [[1, 22, 888, 840], [5, 2, 264, 1464], [865, 770, 24, 1008], [1177, 1418, 888, 144]], 'inner_hash': 4669, 'inner_nilpotent': False, 'inner_order': 864, 'inner_split': False, 'inner_tex': 'C_6^2:D_{12}', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 32, 'irrR_degree': 16, 'irrep_stats': [[1, 4], [2, 9], [3, 4], [4, 9], [6, 1], [8, 6], [12, 4], [16, 2]], 'label': '1728.46260', 'linC_count': 16, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 20, 'linQ_dim': 12, 'linQ_dim_count': 20, 'linR_count': 16, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SL(2,3).SO+(4,2)', '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': 22, 'number_characteristic_subgroups': 17, 'number_conjugacy_classes': 39, 'number_divisions': 33, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 149, 'number_subgroup_classes': 232, 'number_subgroups': 3350, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 19], [3, 80], [4, 348], [6, 224], [8, 144], [12, 624], [24, 288]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 18, 0, 0], 'outer_gens': [[1, 866, 24, 144], [7, 434, 1272, 1344], [1, 1378, 888, 216], [1, 434, 24, 1008]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [288, 121, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [3, 4], [4, 10], [6, 1], [8, 3], [12, 4], [16, 4]], 'representations': {'PC': {'code': 54486844126756658966677087826348652112456471190215154696461110889397493364631894218397080936463199785, 'gens': [1, 2, 5, 7], 'pres': [9, 2, 2, 2, 3, 2, 3, 2, 2, 3, 7776, 397, 46, 542, 74, 579, 39964, 5953, 15142, 7456, 130, 10382, 12983, 1022, 52926, 46131, 12498, 5136, 2688, 186, 110599, 43216, 25945, 214, 27224, 27233, 23354]}, 'Perm': {'d': 22, 'gens': [6403984334840133, 51334588059955200000, 378095736066048000, 10588594971, 4993851490467840000, 104986479713955840000, 2987351515484, 4404319792351, 5808408640626]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SL(2,3).\\SOPlus(4,2)', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
