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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '2304.bc', 'ambient_counter': 29, 'ambient_order': 2304, 'ambient_tex': '(C_6\\times S_4):\\OD_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 144, 'counter': 63, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2304.bc.8.p1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '8.p1', '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': 8, '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': '288.897', 'subgroup_hash': 897, 'subgroup_order': 288, 'subgroup_tex': 'C_{12}\\times S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2304.bc', 'aut_centralizer_order': None, 'aut_label': '8.p1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '96.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1', '4.m1'], 'contains': ['16.b1', '16.p1', '16.s1', '24.eh1', '24.ej1', '32.k1'], 'core': '16.b1', 'coset_action_label': None, 'count': 2, 'diagramx': None, 'generators': [12301, 88211, 90316, 136017, 92011, 11293, 72009], 'label': '2304.bc.8.p1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.c1', 'old_label': '8.p1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.p1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.9', '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], 'aut_gens': [[6209058150, 5710314981, 7341833547, 5249582286, 8157592830, 5711413481, 5743268121], [8561473930, 5710314981, 7341833547, 8678021728, 8157592830, 5711413481, 5743268121], [9686647701, 8973352113, 2447277849, 5249582286, 3263037132, 8973470751, 6552845720], [6209058150, 5710314981, 7341833547, 5249582286, 8157592830, 4898312568, 5743268121], [6209058150, 4894555698, 7341833547, 5249582286, 8157592830, 4898312568, 5743268121], [6209058150, 5710314981, 7341833547, 8678021728, 8157592830, 2203221911, 4260541584], [2049263926, 5710314981, 7341833547, 5249582286, 8157592830, 876082999, 5743268121], [909666381, 5710314981, 7341833547, 5249582286, 8157592830, 876082999, 5743268121]], 'aut_group': '192.1537', 'aut_hash': 1537, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 16, 'aut_perms': [18233886919981, 75247205912, 9693944062369, 9693944063089, 18233331734529, 4058214725, 1311572880240], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 2, 1], [3, 1, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [4, 1, 2, 1], [4, 3, 2, 1], [4, 6, 2, 3], [6, 1, 2, 1], [6, 3, 2, 2], [6, 6, 4, 1], [6, 8, 1, 1], [6, 8, 2, 1], [12, 1, 4, 1], [12, 3, 4, 1], [12, 6, 4, 3], [12, 8, 2, 1], [12, 8, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [2, 6, 2], [3, 1, 2], [3, 8, 3], [4, 1, 2], [4, 3, 2], [4, 6, 6], [6, 1, 2], [6, 3, 4], [6, 6, 4], [6, 8, 3], [12, 1, 4], [12, 3, 4], [12, 6, 12], [12, 8, 6]], 'center_label': '12.2', 'center_order': 12, 'central_product': True, 'central_quotient': '24.12', 'commutator_count': 1, 'commutator_label': '12.3', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 897, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['24.12', 1], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 1, 2], [3, 1, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [4, 1, 2, 1], [4, 3, 2, 1], [4, 6, 1, 2], [4, 6, 2, 2], [6, 1, 2, 1], [6, 3, 2, 2], [6, 6, 2, 2], [6, 8, 1, 1], [6, 8, 2, 1], [12, 1, 4, 1], [12, 3, 4, 1], [12, 6, 2, 2], [12, 6, 4, 2], [12, 8, 2, 1], [12, 8, 4, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 72, 'exponent': 12, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 0, 8]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '144.188', 'hash': 897, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [3, 1, 1, 2, 1, 2, 2], 'inner_gens': [[6209058150, 5710314981, 7341833547, 8678021728, 8157592830, 2203221911, 4260541584], [6209058150, 5710314981, 7341833547, 5249582286, 8157592830, 5711413481, 5743268121], [6209058150, 5710314981, 7341833547, 5249582286, 8157592830, 5711413481, 5743268121], [2049263926, 5710314981, 7341833547, 5249582286, 8157592830, 876082999, 5743268121], [6209058150, 5710314981, 7341833547, 5249582286, 8157592830, 5711413481, 5743268121], [8561473930, 5710314981, 7341833547, 8678021728, 8157592830, 5711413481, 5743268121], [3203925110, 5710314981, 7341833547, 5249582286, 8157592830, 5711413481, 5743268121]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'S_4', 'inner_used': [1, 4, 6], 'irrC_degree': 3, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 6, 'irrep_stats': [[1, 24], [2, 12], [3, 24]], 'label': '288.897', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 40, 'linQ_dim': 7, 'linQ_dim_count': 40, 'linR_count': 16, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C12*S4', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 26, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 60, 'number_divisions': 30, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 104, 'number_subgroup_classes': 118, 'number_subgroups': 374, 'old_label': None, 'order': 288, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 19], [3, 26], [4, 44], [6, 62], [12, 136]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [815759283, 815759283, 815759283], 'outer_gens': [[6209058150, 4894555698, 7341833547, 5249582286, 8157592830, 4898312568, 5743268121], [6209058150, 5710314981, 7341833547, 5249582286, 8157592830, 4898312568, 5743268121], [9686647701, 8973352113, 2447277849, 5249582286, 3263037132, 8973470751, 6552845720]], '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': 4, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [3, 4], [4, 5], [6, 6], [8, 1], [12, 2]], 'representations': {'PC': {'code': 9722728672521459216404654407288962058285833, 'gens': [1, 2, 5, 6], 'pres': [7, -2, -2, -2, -3, -2, 2, -3, 141, 36, 422, 58, 451, 5044, 2531, 1488, 655, 9077, 1524, 2287, 404, 124]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [6209058150, 5710314981, 7341833547, 5249582286, 8157592830, 5711413481, 5743268121]}, 'Perm': {'d': 11, 'gens': [363624, 1704, 3, 744, 403200, 3669120, 7983360]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}\\times S_4', 'transitive_degree': 36, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 6, 6, 4, 6, 6, 4, 12, 12, 6], 'aut_gens': [[12301, 134194, 19633, 88211, 25131], [154219, 114916, 159986, 92011, 91483], [34114, 11509, 88624, 12201, 25131], [156319, 27127, 44550, 88211, 104921], [158219, 130394, 94864, 92011, 28931], [54316, 13209, 102943, 12201, 91483], [54316, 108837, 47670, 12201, 25131], [34114, 8189, 139828, 12201, 95283], [74209, 97398, 120540, 92011, 28931], [78209, 39106, 159796, 92011, 28931], [94211, 157111, 23143, 92011, 15493]], 'aut_group': None, 'aut_hash': 5726325960835554948, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 18432, 'aut_permdeg': 44, 'aut_perms': [512038626667447454351122103769755609705352348863279161, 2607428944714256435557356020247200425585396057149914843, 2084089585124224193759621698198772715567997426088281245, 2349272922637316770759507025991434510201684340686812216, 2274774047121411146107200620525172762958794094687674156, 2244320009258694844655972747943252578831626098502234676, 2619914080524451758711765433128970107193638548067181377, 2228899323199327992772728776998222024169770583047755490, 1337850738598098074328098239140941301546212587223993817, 847800659482987048064730907310338431225087972941541582], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [2, 12, 1, 1], [2, 12, 2, 1], [2, 36, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 1, 2, 2], [4, 3, 2, 2], [4, 12, 1, 1], [4, 12, 2, 3], [4, 36, 1, 1], [4, 72, 1, 4], [6, 2, 1, 3], [6, 6, 1, 4], [6, 8, 1, 3], [6, 16, 1, 3], [6, 24, 2, 1], [6, 48, 2, 1], [8, 4, 2, 1], [8, 6, 4, 1], [8, 12, 2, 1], [8, 18, 4, 1], [8, 24, 2, 2], [8, 36, 4, 2], [12, 2, 2, 2], [12, 6, 2, 2], [12, 8, 2, 2], [12, 16, 2, 2], [12, 24, 2, 3], [12, 48, 2, 1], [24, 4, 4, 1], [24, 12, 4, 1], [24, 16, 4, 1], [24, 16, 8, 1], [24, 24, 4, 2], [24, 48, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_6\\times A_4).C_2^6.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '144.183', 'autcentquo_hash': 183, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [2, 12, 3], [2, 36, 1], [3, 2, 1], [3, 8, 1], [3, 16, 1], [4, 1, 4], [4, 3, 4], [4, 12, 7], [4, 36, 1], [4, 72, 4], [6, 2, 3], [6, 6, 4], [6, 8, 3], [6, 16, 3], [6, 24, 2], [6, 48, 2], [8, 4, 2], [8, 6, 4], [8, 12, 2], [8, 18, 4], [8, 24, 4], [8, 36, 8], [12, 2, 4], [12, 6, 4], [12, 8, 4], [12, 16, 4], [12, 24, 6], [12, 48, 2], [24, 4, 4], [24, 12, 4], [24, 16, 12], [24, 24, 8], [24, 48, 4]], 'center_label': '8.2', 'center_order': 8, 'central_product': False, 'central_quotient': '288.1028', 'commutator_count': 1, 'commutator_label': '144.193', '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'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 29, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [2, 12, 1, 3], [2, 36, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 1, 2, 2], [4, 3, 2, 2], [4, 12, 1, 3], [4, 12, 2, 2], [4, 36, 1, 1], [4, 72, 1, 4], [6, 2, 1, 3], [6, 6, 1, 4], [6, 8, 1, 3], [6, 16, 1, 3], [6, 24, 1, 2], [6, 48, 2, 1], [8, 4, 2, 1], [8, 6, 2, 2], [8, 12, 2, 1], [8, 18, 2, 2], [8, 24, 2, 2], [8, 36, 4, 2], [12, 2, 2, 2], [12, 6, 2, 2], [12, 8, 2, 2], [12, 16, 2, 2], [12, 24, 1, 2], [12, 24, 2, 2], [12, 48, 2, 1], [24, 4, 4, 1], [24, 12, 4, 1], [24, 16, 4, 1], [24, 16, 8, 1], [24, 24, 4, 2], [24, 48, 2, 2]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 161280, 'exponent': 24, 'exponents_of_order': [8, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '288.1028', 'hash': 1492703109563463400, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 4, 6, 2, 6], 'inner_gens': [[12301, 10089, 59828, 88211, 104921], [10201, 134194, 24792, 92011, 91483], [14201, 152031, 19633, 92011, 15493], [12301, 54184, 99823, 88211, 25131], [92311, 49304, 65467, 88211, 25131]], 'inner_hash': 1028, 'inner_nilpotent': False, 'inner_order': 288, 'inner_split': True, 'inner_tex': 'D_6\\times S_4', 'inner_used': [1, 2, 3, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 44], [3, 16], [4, 24], [6, 28], [12, 4]], 'label': '2304.bc', '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*S4):OD16', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 69, 'number_characteristic_subgroups': 93, 'number_conjugacy_classes': 132, 'number_divisions': 76, 'number_normal_subgroups': 103, 'number_subgroup_autclasses': 1160, 'number_subgroup_classes': 1356, 'number_subgroups': 11176, 'old_label': None, 'order': 2304, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 87], [3, 26], [4, 424], [6, 246], [8, 512], [12, 368], [24, 640]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2, 2], 'outer_gen_pows': [8001, 8001, 8001, 8001, 8001, 8001], 'outer_gens': [[94211, 54184, 143838, 92011, 28931], [12301, 134194, 19633, 88211, 95283], [158219, 54184, 143838, 92011, 28931], [94211, 54184, 108782, 92011, 91483], [94211, 54184, 59638, 92011, 28931], [94211, 54184, 10874, 92011, 91483]], 'outer_group': '64.267', 'outer_hash': 267, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 12, 'outer_perms': [39916800, 362880, 5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [3, 8], [4, 13], [6, 10], [8, 10], [12, 7], [16, 2], [24, 4]], 'representations': {'PC': {'code': '73355782718668104112677712730521292380966249717231059008648265861592188661831550582437167600410235980575654136095468527638342665', 'gens': [1, 2, 4, 8, 9], 'pres': [10, -2, -2, -2, -2, -2, -2, -3, -2, 2, -3, 1920, 6521, 51, 5882, 59203, 27693, 11623, 113, 13604, 6814, 2624, 144, 90245, 45135, 17785, 175, 22406, 11216, 4506, 46097, 11557, 6767, 2937, 1747, 51848, 43218, 23798, 9768, 1678, 2498, 268, 76819, 19239]}, 'GLZN': {'d': 2, 'p': 20, 'gens': [11293, 24003, 103948, 72009, 8189, 60207, 130106, 130311, 136017, 8201]}, 'Perm': {'d': 19, 'gens': [4, 186834697200, 6402467594488464, 8361600, 6779077809159865, 6423483318751680, 19918583639030400, 19562808545280000, 12511680, 13516309078272000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_6\\times S_4):\\OD_{16}', 'transitive_degree': 288, 'wreath_data': None, 'wreath_product': False}
