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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '960.1058', 'ambient_counter': 1058, 'ambient_order': 960, 'ambient_tex': 'C_{60}.\\OD_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 24, 'counter': 79, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.1058.20.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '20.f1.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': 20, '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': '48.23', 'subgroup_hash': 23, 'subgroup_order': 48, 'subgroup_tex': 'C_2\\times C_{24}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.1058', 'aut_centralizer_order': 384, 'aut_label': '20.f1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '16.14', 'aut_weyl_index': 1920, 'centralizer': '10.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.c1.a1', '10.b1.a1', '10.d1.a1', '10.d1.b1'], 'contains': ['40.a1.a1', '40.g1.a1', '40.g1.b1', '60.c1.a1'], 'core': '40.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [9924, -1, 9944, -1, 6632, -1, 6852, -1], 'generators': [1, 520, 480, 320, 2], 'label': '960.1058.20.f1.a1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '20.f1.a1', 'projective_image': '120.42', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '20.f1.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': '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, 4], 'aut_gens': [[1, 2], [1, 14], [1, 22], [25, 22], [25, 15]], 'aut_group': '32.46', 'aut_hash': 46, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [16, 23, 11536, 16576], '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, 2, 2], [6, 1, 2, 1], [6, 1, 4, 1], [8, 1, 8, 1], [12, 1, 4, 2], [24, 1, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.46', 'autcent_hash': 46, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\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], [8, 1, 8], [12, 1, 8], [24, 1, 16]], 'center_label': '48.23', 'center_order': 48, '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', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 23, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['8.1', 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], [8, 1, 4, 2], [12, 1, 4, 2], [24, 1, 8, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 24, 'exponent': 24, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '12.5', 'hash': 23, '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, 48]], 'label': '48.23', 'linC_count': 384, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C24', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 48, 'number_divisions': 16, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 22, 'number_subgroups': 22, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [8, 8], [12, 8], [24, 16]], '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], [1, 22], [25, 22], [25, 15]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [16, 23, 11536, 16576], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 8, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 4], [8, 2]], 'representations': {'PC': {'code': 1286603201, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -2, -3, 26, 42, 58]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [24992906222183252, 41624324273564377]}, 'GLFp': {'d': 2, 'p': 73, 'gens': [17894809, 20228936]}, 'Perm': {'d': 13, 'gens': [27760320, 479001600, 4, 11614560, 3669864]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 24], '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_{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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [20, 2, 4, 2, 4, 10, 4, 10, 4], 'aut_gens': [[1, 4, 80], [73, 764, 600], [67, 36, 560], [43, 764, 120], [3, 556, 80], [35, 748, 880], [51, 164, 440], [523, 764, 120], [17, 324, 400], [539, 908, 920]], 'aut_group': None, 'aut_hash': 2878935608206837536, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 30720, 'aut_permdeg': 64, 'aut_perms': [72096152218319763194413846335622169701432849974793164791973246443561082158965728107636547, 46230422833086147762410872437587317628062536876789288299703730394799871928179860572759638, 86698219863061864463949956772307108253834929313722702902941653255776428326748134082778930, 24503764658780828249480004638198867268001226556493649846773380391420766131413810935071318, 34426602524651933966846817428265162148085587922797502804512974260845833892804059733495946, 40348650841520989179326355389051702079536302338644405124268636477483376108770120043606972, 98719749662518242584394358096243375466381810285806558114373152153056910186720604202561330, 2109511833370973523519373713789273281493130702348465111600554567367782912668274688226186, 100578034516203641607522605554584411177698184837078290254324176112913859562158956642217599], 'aut_phi_ratio': 120.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 1, 2, 2], [4, 2, 2, 2], [4, 12, 2, 2], [5, 2, 2, 1], [6, 2, 1, 3], [8, 10, 4, 2], [8, 60, 4, 1], [10, 2, 2, 3], [12, 2, 2, 2], [12, 2, 4, 2], [15, 4, 2, 1], [20, 2, 4, 2], [20, 4, 4, 2], [20, 12, 8, 2], [24, 10, 8, 2], [30, 4, 2, 3], [60, 4, 4, 2], [60, 4, 8, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_{15}:(C_2^4\\times C_4\\times C_2^2\\wr C_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': 60, 'autcentquo_group': '240.195', 'autcentquo_hash': 195, 'autcentquo_nilpotent': False, 'autcentquo_order': 240, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 1], [4, 1, 4], [4, 2, 4], [4, 12, 4], [5, 2, 2], [6, 2, 3], [8, 10, 8], [8, 60, 4], [10, 2, 6], [12, 2, 12], [15, 4, 2], [20, 2, 8], [20, 4, 8], [20, 12, 16], [24, 10, 16], [30, 4, 6], [60, 4, 24]], 'center_label': '8.2', 'center_order': 8, 'central_product': False, 'central_quotient': '120.42', 'commutator_count': 1, 'commutator_label': '60.13', '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', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 1058, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 12, 1, 4], [5, 2, 2, 1], [6, 2, 1, 3], [8, 10, 2, 2], [8, 10, 4, 1], [8, 60, 2, 2], [10, 2, 2, 3], [12, 2, 2, 4], [12, 2, 4, 1], [15, 4, 2, 1], [20, 2, 4, 2], [20, 4, 2, 2], [20, 4, 4, 1], [20, 12, 4, 4], [24, 10, 4, 2], [24, 10, 8, 1], [30, 4, 2, 3], [60, 4, 4, 4], [60, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 8064, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '120.42', 'hash': 1058, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 10, 6], 'inner_gens': [[1, 556, 80], [489, 4, 880], [1, 164, 80]], 'inner_hash': 42, 'inner_nilpotent': False, 'inner_order': 120, 'inner_split': False, 'inner_tex': 'S_3\\times D_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 76], [4, 40]], 'label': '960.1058', 'linC_count': 256, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 160, 'linQ_dim': 14, 'linQ_dim_count': 104, 'linR_count': 176, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C60.OD16', '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': 41, 'number_characteristic_subgroups': 58, 'number_conjugacy_classes': 132, 'number_divisions': 52, 'number_normal_subgroups': 92, 'number_subgroup_autclasses': 136, 'number_subgroup_classes': 188, 'number_subgroups': 616, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 60], [5, 4], [6, 6], [8, 320], [10, 12], [12, 24], [15, 8], [20, 240], [24, 160], [30, 24], [60, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 48, 0, 33, 240, 0, 640], 'outer_gens': [[65, 876, 880], [531, 324, 880], [1, 364, 400], [491, 468, 440], [491, 956, 400], [67, 36, 560], [49, 868, 560]], 'outer_group': '256.56085', 'outer_hash': 56085, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 128, 'outer_perms': [351097902750379604218486221534484697977980164010593284793763401767121270741849152361175010791362963956842362486544990890745066915982850439711646269256295537497286396201654513108650979666294660912983445373938531084036, 302133512315038466322249626284334359905607566634712311383539666094550863928911153778598834926067187127066559363806920463754677662994837171763009402988293223455620632379547546430920557492303513664828706923779857924808, 121731523049591757171399962139285317322843057659047787546316948655353238493651211617696523445608431881356831924891299817872011684398648711918458623654692730712323672652263800153674375838813103965751548899721629632868, 181809100511571472789690411843685642329547132702322374479106986170123997770082255345924305065334414098367577512589561143962057318663860870788886683379475217974307862713989164922081166442799051154172210272124834831379, 236098172543862600973041185399884788058004621228876616121236372942085812193580332051018660836512151169710172517607476095497229672951564092158983607512734058086767144332170203848129452673748732801143035922561535973166, 6238830849388261225738111787484024626780325783776521903604353357026646180800595727352113773127806438126586706132302321160847682719802689001529260778371385374846014554555632571243417769407542888808407968214617318908, 328063424788676862063022317789069773867161228831717172978372823051994988536378051918600362961743347752722030713296533256096517590746074053319531652684614391255553082175019445897187448955988679782510430427941623930146], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4:C_2^5', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 15], [8, 10], [16, 8], [32, 1]], 'representations': {'PC': {'code': 1440189201435277658658244178995965342911233458369, 'gens': [1, 3, 6], 'pres': [8, -2, -2, -2, -2, -5, -2, -2, -3, 16, 4169, 13346, 66, 2307, 91, 2564, 10581, 141, 11222, 166, 10263]}, 'Perm': {'d': 24, 'gens': [28207315932383002424424, 3425539190832582566401, 55083347176171961162880, 82261660440517189632000, 1321086775680, 55083347171874717696000, 3, 6504]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{60}.\\OD_{16}', 'transitive_degree': 960, 'wreath_data': None, 'wreath_product': False}
