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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '3840.hv', 'ambient_counter': 204, 'ambient_order': 3840, 'ambient_tex': 'D_{10}.D_4\\times S_4', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 2, 'counter': 1275, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '3840.hv.320.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '320.f1', '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': 320, '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': '12.4', 'subgroup_hash': 4, 'subgroup_order': 12, 'subgroup_tex': 'D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3840.hv', 'aut_centralizer_order': 256, 'aut_label': '320.f1', 'aut_quo_index': None, 'aut_stab_index': 40, 'aut_weyl_group': '12.4', 'aut_weyl_index': 10240, 'centralizer': '120.br1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['64.f1', '80.q1', '160.f1', '160.k1', '160.p1', '160.u1', '160.v1'], 'contains': ['640.a1', '640.g1', '960.k1'], 'core': '1920.a1', 'coset_action_label': None, 'count': 40, 'diagramx': None, 'generators': [87664953600, 11249280, 87664917058], 'label': '3840.hv.320.f1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '16.e1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '20.j1', 'old_label': '320.f1', 'projective_image': '1920.240396', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '320.f1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [1, 10], [3, 2]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 2, 1], [6, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 10], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.4', 'linC_count': 1, 'linC_degree': 2, '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': 'D6', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 16, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[7, 2]], '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': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 43377, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 61, 16, 74]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [17, 35]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 55, 56]}, 'Perm': {'d': 5, 'gens': [6, 1, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_6', 'transitive_degree': 6, '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': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [12, 12, 4, 12, 60, 4, 4], 'aut_gens': [[11652480, 187290393360, 187296883205, 87657292800, 3669120, 187289625600, 186815019749, 7620480, 7660800, 187293298706], [3669120, 187289989282, 187300874885, 87657292800, 7983360, 187289625600, 486261152, 11289605, 10926720, 187297611440], [7983360, 187289666720, 187296883205, 87657292800, 11652480, 273988915200, 274468321952, 87660921600, 10926720, 274000568882], [7983360, 187300916000, 273992906880, 87657292800, 3669120, 187289625605, 87189219872, 362885, 403200, 187293296200], [3669120, 187300915336, 273989318400, 87657292800, 7983360, 187289625605, 186815019200, 362880, 3991680, 187297610440], [3669120, 273989278214, 187294020485, 87657292800, 7983360, 187289625600, 479405122, 87668582400, 7660800, 187297612946], [3669120, 273996537365, 273999841925, 87657292800, 11652480, 273988915200, 186818285669, 766085, 403200, 274000569158], [11652480, 273996538370, 187290028800, 87657292800, 3669120, 273988915205, 186821876908, 766085, 10926720, 273992585800]], 'aut_group': None, 'aut_hash': 6217586336571907406, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 122880, 'aut_permdeg': 64, 'aut_perms': [13011735837380969176986527137836000555296392834781832694403372950835146835349010856271300, 101647685584462085466250113476159271871359141908572445399282969640998480907792734383853495, 12046320587671345287933917212664167949664911882559356103673229938079982156259398121496954, 58510116679005602971089348576679592046090354887231960207199599178978043902124621456211970, 40020629866250355395136734627543806683830558237735535232868325863767428696782898199177167, 9062362166168800044822148615378834348586323976773415573435298717141580525824892433157359, 4131022602204260285763646483187170239370262035367399994184451256198113108594312850130916], 'aut_phi_ratio': 120.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 5, 1, 2], [2, 5, 2, 1], [2, 6, 4, 1], [2, 15, 1, 2], [2, 15, 2, 1], [2, 30, 4, 1], [3, 8, 1, 1], [4, 2, 2, 1], [4, 6, 2, 1], [4, 6, 4, 1], [4, 10, 2, 1], [4, 10, 4, 2], [4, 12, 2, 2], [4, 30, 2, 1], [4, 30, 4, 3], [4, 60, 2, 2], [4, 60, 4, 4], [5, 4, 1, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 40, 1, 2], [6, 40, 2, 1], [10, 4, 1, 1], [10, 4, 2, 1], [10, 12, 1, 2], [10, 12, 2, 1], [10, 24, 4, 1], [12, 16, 2, 1], [12, 80, 2, 1], [12, 80, 4, 2], [15, 32, 1, 1], [20, 4, 4, 1], [20, 12, 4, 1], [20, 24, 4, 3], [30, 32, 1, 1], [30, 32, 2, 1], [60, 32, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_5\\times A_4).C_2^6.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '960.11361', 'autcentquo_hash': 11361, 'autcentquo_nilpotent': False, 'autcentquo_order': 960, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times F_5\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [2, 5, 4], [2, 6, 4], [2, 15, 4], [2, 30, 4], [3, 8, 1], [4, 2, 2], [4, 6, 6], [4, 10, 10], [4, 12, 4], [4, 30, 14], [4, 60, 20], [5, 4, 1], [6, 8, 3], [6, 40, 4], [10, 4, 3], [10, 12, 4], [10, 24, 4], [12, 16, 2], [12, 80, 10], [15, 32, 1], [20, 4, 4], [20, 12, 4], [20, 24, 12], [30, 32, 3], [60, 32, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '960.11361', 'commutator_count': 1, 'commutator_label': '120.43', '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', '5.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 204, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['24.12', 1], ['80.31', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [2, 5, 1, 4], [2, 6, 1, 4], [2, 15, 1, 4], [2, 30, 1, 4], [3, 8, 1, 1], [4, 2, 1, 2], [4, 6, 1, 6], [4, 10, 1, 2], [4, 10, 2, 4], [4, 12, 1, 4], [4, 30, 1, 6], [4, 30, 2, 4], [4, 60, 1, 4], [4, 60, 2, 8], [5, 4, 1, 1], [6, 8, 1, 3], [6, 40, 1, 4], [10, 4, 1, 3], [10, 12, 1, 4], [10, 24, 1, 4], [12, 16, 1, 2], [12, 80, 1, 2], [12, 80, 2, 4], [15, 32, 1, 1], [20, 4, 2, 2], [20, 12, 2, 2], [20, 24, 1, 4], [20, 24, 2, 4], [30, 32, 1, 3], [60, 32, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 511842240, 'exponent': 60, 'exponents_of_order': [8, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '1920.240396', 'hash': 2337493134112712028, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 2, 6, 1, 2, 2, 12, 2, 3, 10], 'inner_gens': [[11652480, 187297247760, 187290028805, 87657292800, 3669120, 187289625600, 186821874149, 766080, 10926720, 187293298706], [3669120, 187290393360, 187297286405, 87657292800, 11652480, 187289625600, 274478844389, 7620480, 7257600, 187301280560], [3669120, 187300916880, 187296883205, 87657292800, 7983360, 187289625600, 274479166949, 362880, 725760, 187297612946], [11652480, 187290393360, 187296883205, 87657292800, 3669120, 187289625600, 186815019749, 7620480, 7660800, 187293298706], [11652480, 187297247760, 187294020485, 87657292800, 3669120, 187289625600, 186817882469, 766080, 725760, 187293298706], [11652480, 187290393360, 187296883205, 87657292800, 3669120, 187289625600, 274472312549, 7620480, 7660800, 187293298706], [3669120, 274000206480, 273989318405, 87657292800, 7983360, 273988915200, 186815019749, 362880, 3991680, 273996899762], [3669120, 187290393360, 187293617285, 87657292800, 11652480, 187289625600, 186811350629, 7620480, 11249280, 187301282066], [7983360, 187293256080, 187290028805, 87657292800, 11652480, 187289625600, 186817882469, 40320, 7660800, 187301282066], [11652480, 187297248770, 187294020485, 87657292800, 3669120, 187289625600, 274475178028, 766080, 725760, 187293298706]], 'inner_hash': 11361, 'inner_nilpotent': False, 'inner_order': 960, 'inner_split': False, 'inner_tex': 'C_2\\times F_5\\times S_4', 'inner_used': [1, 2, 3, 7, 10], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [3, 32], [4, 20], [6, 8], [8, 8], [12, 16]], 'label': '3840.hv', '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': None, 'name': 'D10.D4*S4', '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, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 58, 'number_characteristic_subgroups': 64, 'number_conjugacy_classes': 140, 'number_divisions': 110, 'number_normal_subgroups': 288, 'number_subgroup_autclasses': 1538, 'number_subgroup_classes': 4786, 'number_subgroups': 58388, 'old_label': None, 'order': 3840, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 239], [3, 8], [4, 1808], [5, 4], [6, 184], [10, 156], [12, 832], [15, 32], [20, 352], [30, 96], [60, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 2, 4], 'outer_gen_pows': [0, 1680, 0, 1680, 0], 'outer_gens': [[11652480, 273989682960, 273996172800, 87657292800, 3669120, 273988915205, 274472312544, 7620485, 7660800, 273992588308], [11652480, 187290393365, 187296883205, 87657292800, 3669120, 187289625600, 274472312544, 7620485, 7660800, 187293296198], [11652480, 187290393365, 273996172800, 87657292800, 3669120, 273988915205, 87182686944, 87664913280, 7660800, 273992585800], [11652480, 273989682960, 187296883205, 87657292800, 3669120, 187289625600, 274472312544, 87664913280, 7660800, 187293296198], [11652480, 273989682965, 273996172805, 87657292800, 3669120, 187289625600, 483397349, 87664913285, 7660800, 187293296198]], 'outer_group': '128.1135', 'outer_hash': 1135, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [167903, 40805407, 90882862, 167450527, 135814321], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 24], [3, 16], [4, 16], [6, 16], [8, 8], [12, 8], [16, 2], [24, 4]], 'representations': {'PC': {'code': '9345060256162749240612229612712812274024094810393666223238355357454351348659985558016012', 'gens': [1, 2, 5, 7, 8, 10], 'pres': [10, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 480, 1161, 51, 602, 82, 643, 1814, 144, 295687, 101777, 53787, 21637, 237, 138248, 51858, 34588, 48019, 73229, 12039]}, 'Perm': {'d': 15, 'gens': [11652480, 187290393360, 187296883205, 87657292800, 3669120, 187289625600, 186815019749, 7620480, 7660800, 187293298706]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_{10}.D_4\\times S_4', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
