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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.922', 'ambient_counter': 922, 'ambient_order': 1344, 'ambient_tex': 'C_{28}.(C_6\\times D_4)', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 32, 'counter': 85, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.922.14.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '14.c1.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': 14, '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': '96.48', 'subgroup_hash': 48, 'subgroup_order': 96, 'subgroup_tex': 'C_2^2:C_{24}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.922', 'aut_centralizer_order': 48, 'aut_label': '14.c1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '32.51', 'aut_weyl_index': 336, 'centralizer': '56.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1', '7.a1.a1'], 'contains': ['28.b1.a1', '28.i1.a1', '28.j1.a1', '42.a1.a1'], 'core': '42.a1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [1055, -1, 9590, -1, 8260, -1, 1302, -1], 'generators': [6, 24, 48, 672, 16, 12], 'label': '1344.922.14.c1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '14.c1.a1', 'projective_image': '336.123', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '14.c1.a1', 'subgroup_fusion': None, 'weyl_group': '8.5'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '48.23', '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, 2, 4], 'aut_gens': [[1, 2, 16], [49, 14, 16], [49, 54, 80], [57, 6, 80], [9, 6, 80], [1, 10, 16], [9, 55, 16]], 'aut_group': '128.2163', 'aut_hash': 2163, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 128, 'aut_permdeg': 12, 'aut_perms': [5176, 16680, 7637160, 7632023, 87091200, 119755569], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [3, 1, 2, 1], [4, 1, 4, 1], [4, 2, 2, 1], [6, 1, 2, 3], [6, 2, 4, 1], [8, 2, 8, 1], [12, 1, 8, 1], [12, 2, 4, 1], [24, 2, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [3, 1, 2], [4, 1, 4], [4, 2, 2], [6, 1, 6], [6, 2, 4], [8, 2, 8], [12, 1, 8], [12, 2, 4], [24, 2, 16]], 'center_label': '24.9', 'center_order': 24, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 48, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['32.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [3, 1, 2, 1], [4, 1, 2, 2], [4, 2, 2, 1], [6, 1, 2, 3], [6, 2, 2, 2], [8, 2, 4, 2], [12, 1, 4, 2], [12, 2, 4, 1], [24, 2, 8, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 24, 'exponent': 24, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '12.5', 'hash': 48, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1], 'inner_gens': [[1, 50, 16], [49, 2, 16], [1, 2, 16]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48], [2, 12]], 'label': '96.48', 'linC_count': 256, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 30, 'linQ_dim': 8, 'linQ_dim_count': 30, 'linR_count': 16, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2:C24', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 16, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 60, 'number_divisions': 22, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 38, 'number_subgroup_classes': 50, 'number_subgroups': 68, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 8], [6, 14], [8, 16], [12, 16], [24, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 14, 16], [1, 14, 80], [9, 14, 80], [9, 15, 16]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [7, 16, 1560, 5896], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 8, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 7], [8, 3]], 'representations': {'PC': {'code': 238152446905729, 'gens': [1, 2, 5], 'pres': [6, -2, -2, -2, -2, -2, -3, 601, 31, 50, 88]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 92135121190, 79209322247, 23025133754, 61054693752, 103603286778]}, 'Perm': {'d': 15, 'gens': [93888760464, 87665654400, 4, 186810624000, 186823135680, 8361600]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 24], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2:C_{24}', 'transitive_degree': 48, '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 2, 6, 14, 6, 12, 6, 6], 'aut_gens': [[1, 2, 48, 96], [73, 62, 48, 912], [1201, 1034, 48, 912], [841, 194, 48, 1272], [805, 230, 48, 1056], [277, 470, 48, 144], [193, 422, 48, 528], [613, 1226, 48, 360], [769, 206, 48, 504], [793, 254, 48, 1128]], 'aut_group': None, 'aut_hash': 5229004673012084454, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10752, 'aut_permdeg': 68, 'aut_perms': [1252211669945184698914662494880803307641557648364387725860278601030299286677463673584303548130439, 666931705365590514899620651667001631088366884524556405431265110129948322820960746085318375535556, 784668842619280437345117575185162989348875990990036051198822799605801078300838698693426529281128, 2445197720164615810015241137424394983045431982894764274569197687426721163779912755681759371286813, 37023949469073462624891698257907294058270298171370213233346947926479498052196157128526996379453, 931088867006695969894603957697669317732884417483406982389281290539103457347092995556358376407797, 1625766431490878051970995190683029441696493987993586233746736672106933183536799416700798649967169, 1921890475458717582716174135987301933636008437370897923617233696260155183577890819633808349598132, 1447740897068207903793064257399715703223437698913633917696217665270533010590499367668071462490212], 'aut_phi_ratio': 28.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 28, 2, 1], [3, 7, 1, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 28, 2, 1], [4, 56, 1, 2], [6, 7, 1, 6], [6, 28, 1, 2], [6, 28, 2, 2], [7, 6, 1, 1], [8, 4, 2, 2], [12, 14, 1, 4], [12, 14, 2, 2], [12, 28, 2, 2], [12, 56, 1, 4], [14, 6, 1, 3], [14, 12, 2, 1], [24, 28, 2, 4], [28, 6, 2, 2], [28, 12, 2, 1], [56, 12, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_7.(C_2^5\\times C_6).C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 1], [2, 28, 2], [3, 7, 2], [4, 2, 4], [4, 28, 2], [4, 56, 2], [6, 7, 6], [6, 28, 6], [7, 6, 1], [8, 4, 4], [12, 14, 8], [12, 28, 4], [12, 56, 4], [14, 6, 3], [14, 12, 2], [24, 28, 8], [28, 6, 4], [28, 12, 2], [56, 12, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '336.123', 'commutator_count': 1, 'commutator_label': '56.8', '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', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 922, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 28, 1, 2], [3, 7, 2, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 28, 1, 2], [4, 56, 1, 2], [6, 7, 2, 3], [6, 28, 2, 3], [7, 6, 1, 1], [8, 4, 2, 2], [12, 14, 2, 2], [12, 14, 4, 1], [12, 28, 2, 2], [12, 56, 2, 2], [14, 6, 1, 3], [14, 12, 2, 1], [24, 28, 4, 2], [28, 6, 2, 2], [28, 12, 2, 1], [56, 12, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 69888, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '168.47', 'hash': 922, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 12, 1, 14], 'inner_gens': [[1, 38, 48, 1272], [13, 2, 48, 1104], [1, 2, 48, 96], [217, 434, 48, 96]], 'inner_hash': 123, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': None, 'inner_tex': 'D_{28}:C_6', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 30], [4, 3], [6, 16], [12, 4]], 'label': '1344.922', 'linC_count': 96, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 16, 'linQ_dim': 16, 'linQ_dim_count': 48, 'linR_count': 192, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C28.(C6*D4)', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 52, 'number_characteristic_subgroups': 64, 'number_conjugacy_classes': 77, 'number_divisions': 42, 'number_normal_subgroups': 72, 'number_subgroup_autclasses': 244, 'number_subgroup_classes': 304, 'number_subgroups': 1988, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 63], [3, 14], [4, 176], [6, 210], [7, 6], [8, 16], [12, 448], [14, 42], [24, 224], [28, 48], [56, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2], 'outer_gen_pows': [32, 416, 0, 592, 608], 'outer_gens': [[73, 62, 48, 912], [1201, 1034, 48, 912], [841, 194, 48, 1272], [805, 230, 48, 1056], [193, 422, 48, 528]], 'outer_group': '32.51', 'outer_hash': 51, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 32, 'outer_perms': [8286944364775336775771234688974663, 18571810710538201380391455935862960, 27179691438092070656247384601278720, 35679870403808096255989731269909896, 44178822125969133285376195933751740], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 7], [6, 4], [8, 2], [12, 4], [16, 1], [24, 1], [48, 1]], 'representations': {'PC': {'code': 4881353116382799530003615173439531714874570383191867705791647453, 'gens': [1, 2, 6, 7], 'pres': [8, 2, 2, 2, 2, 3, 2, 2, 7, 609, 41, 674, 66, 91, 71238, 30926, 12118, 7422, 3062, 166, 73735, 24591, 6167, 6175, 1575]}, 'Perm': {'d': 27, 'gens': [405964590934781901000343807, 886775988056277929791870080, 1307370579266313975644582400, 304, 7983360, 1690932141767855827749807360, 2127752503579100291781888000, 978]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_{28}.(C_6\\times D_4)', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
