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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1376.36', 'ambient_counter': 36, 'ambient_order': 1376, 'ambient_tex': 'C_2^2\\times C_{344}', 'central': True, 'central_factor': False, 'centralizer_order': 1376, 'characteristic': True, 'core_order': 32, 'counter': 39, 'cyclic': False, 'direct': True, 'hall': 2, 'label': '1376.36.43.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '43.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '43.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 43, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{43}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '32.36', 'subgroup_hash': 36, 'subgroup_order': 32, 'subgroup_tex': 'C_2^2\\times C_8', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '1376.36', 'aut_centralizer_order': 42, 'aut_label': '43.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '384.20089', 'aut_weyl_index': 42, 'centralizer': '1.a1.a1', 'complements': ['32.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['86.a1.a1', '86.a1.b1', '86.a1.c1', '86.a1.d1', '86.a1.e1', '86.a1.f1', '86.b1.a1'], 'core': '43.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [9013, 9013, 1627, 1627, 6648, 6648, 8874, 8874], 'generators': [1, 2, 172], 'label': '1376.36.43.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '43.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['86.a1.b1', '86.a1.a1', '86.a1.d1', '86.a1.c1', '86.a1.e1', '86.a1.f1', '86.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '43.a1.a1', 'projective_image': '43.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '43.1', 'quotient_action_kernel_order': 43, 'quotient_fusion': None, 'short_label': '43.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [3, 2, 4], [1, 2, 28], [2, 3, 4], [1, 2, 6], [1, 2, 7], [1, 18, 4], [17, 2, 4], [1, 2, 20]], 'aut_group': '384.20089', 'aut_hash': 20089, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 10, 'aut_perms': [374407, 1, 823080, 2810574, 2003886, 367920, 5160, 368046], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [4, 1, 2, 1], [4, 1, 6, 1], [8, 1, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '384.20089', 'autcent_hash': 20089, 'autcent_nilpotent': False, 'autcent_order': 384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4:S_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, 7], [4, 1, 8], [8, 1, 16]], 'center_label': '32.36', 'center_order': 32, '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', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 36, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 1, 2, 4], [8, 1, 4, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 28, 'exponent': 8, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '8.5', 'hash': 36, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 32]], 'label': '32.36', 'linC_count': 1792, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 48, 'linQ_dim': 6, 'linQ_dim_count': 48, 'linR_count': 96, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C8', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 32, 'number_divisions': 16, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 38, 'number_subgroups': 38, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 8], [8, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[3, 2, 4], [1, 2, 28], [2, 3, 4], [1, 2, 6], [1, 2, 7], [1, 18, 4], [17, 2, 4], [1, 2, 20]], 'outer_group': '384.20089', 'outer_hash': 20089, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 10, 'outer_perms': [374407, 1, 823080, 2810574, 2003886, 367920, 5160, 368046], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 4]], 'representations': {'PC': {'code': 557068, 'gens': [1, 2, 3], 'pres': [5, -2, 2, 2, -2, -2, 42, 58]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [25038659807093174, 125101739941717066, 24992906222183252]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [34876415, 3397304, 28816400, 3054512, 28233362]}, 'Perm': {'d': 12, 'gens': [40176, 362880, 39916800, 16582, 5167]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_8', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '1376.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 84, 'aut_gen_orders': [84, 6, 28, 28], 'aut_gens': [[1, 2, 4], [689, 690, 1101], [1, 690, 836], [1, 691, 1214], [3, 690, 110]], 'aut_group': None, 'aut_hash': 6908974570931483009, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 58, 'aut_perms': [2220672189599751157261724677988890735565583036969196714782096887255591377493656, 28717181174644529847745885481947942965180266949573312480985132984193752880494, 1488168488202635480170018207512658212997808485002622144565971205938646300499565, 2117375818355270330365022565951039708371370371357522537861315175337472236629367], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [4, 1, 2, 1], [4, 1, 6, 1], [8, 1, 16, 1], [43, 1, 42, 1], [86, 1, 42, 1], [86, 1, 252, 1], [172, 1, 84, 1], [172, 1, 252, 1], [344, 1, 672, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3:A_4.C_{42}.C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': None, 'autcent_hash': 6908974570931483009, 'autcent_nilpotent': False, 'autcent_order': 16128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3:A_4.C_{42}.C_2^2', '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, 7], [4, 1, 8], [8, 1, 16], [43, 1, 42], [86, 1, 294], [172, 1, 336], [344, 1, 672]], 'center_label': '1376.36', 'center_order': 1376, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '43.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 36, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['43.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 1, 2, 4], [8, 1, 4, 4], [43, 1, 42, 1], [86, 1, 42, 7], [172, 1, 84, 4], [344, 1, 168, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 53004, 'exponent': 344, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 43], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '344.12', 'hash': 36, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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': None, 'irrep_stats': [[1, 1376]], 'label': '1376.36', 'linC_count': None, 'linC_degree': 3, '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C344', 'ngens': 6, 'nilpotency_class': 1, '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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 1376, 'number_divisions': 32, 'number_normal_subgroups': 76, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 76, 'number_subgroups': 76, 'old_label': None, 'order': 1376, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 8], [8, 16], [43, 42], [86, 294], [172, 336], [344, 672]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [42, 28, 42, 84], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[691, 689, 983], [689, 3, 188], [2, 691, 620], [691, 2, 813]], 'outer_group': None, 'outer_hash': 6908974570931483009, 'outer_nilpotent': False, 'outer_order': 16128, 'outer_permdeg': 58, 'outer_perms': [1596043582149381907301206559701289059691704463425145339139541929479971346043324, 411536235419030602990280578383118149470886732244967423786126136103059012534683, 1034498712358187291522210485369129437513336188055376565867752281030257754890161, 2192609969757232188998479618121551388268891342550708517875791622357175748011246], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:A_4.C_{42}.C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 55, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 8, 43], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 4], [42, 8], [84, 4], [168, 4]], 'representations': {'PC': {'code': 11238687956725444599593, 'gens': [1, 2, 3], 'pres': [6, -2, -2, -2, -2, -2, -43, 50, 69, 88]}, 'Perm': {'d': 55, 'gens': [216599033751535440633927791155991262120042770706947112960000000000, 80658175170943878571660636856403766975289505440883277824000000000000, 230843697339241380472092742683027581083278564571807941132288000000000000, 92458860080319590419809567499679506678988139383362682880000000000, 30426512622586345503160869073461196830685671907286581248000000000, 59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 344], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{344}', 'transitive_degree': 1376, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '43.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 42, 'aut_gen_orders': [42], 'aut_gens': [[1], [3]], 'aut_group': '42.6', 'aut_hash': 6, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 42, 'aut_permdeg': 12, 'aut_perms': [40320873], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [43, 1, 42, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 42, 'autcent_group': '42.6', 'autcent_hash': 6, 'autcent_nilpotent': True, 'autcent_order': 42, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{42}', '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], [43, 1, 42]], 'center_label': '43.1', 'center_order': 43, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['43.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [43, 1, 42, 1]], 'element_repr_type': 'PC', 'elementary': 43, 'eulerian_function': 1, 'exponent': 43, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [43], 'faithful_reps': [[1, 0, 42]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '43.1', 'hash': 1, 'hyperelementary': 43, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 42, 'irrQ_dim': 42, 'irrR_degree': 2, 'irrep_stats': [[1, 43]], 'label': '43.1', 'linC_count': 42, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 42, 'linQ_degree_count': 1, 'linQ_dim': 42, 'linQ_dim_count': 1, 'linR_count': 21, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C43', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 43, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 43, 'order_factorization_type': 1, 'order_stats': [[1, 1], [43, 42]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 42, 'outer_gen_orders': [42], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '42.6', 'outer_hash': 6, 'outer_nilpotent': True, 'outer_order': 42, 'outer_permdeg': 12, 'outer_perms': [40320873], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{42}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 43, 'pgroup': 43, 'primary_abelian_invariants': [43], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [42, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -43]}, 'Lie': [{'d': 1, 'q': 43, 'gens': [1439295494700374021157505910939096377494040420940313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 43, 'gens': [79551]}, 'Perm': {'d': 43, 'gens': [59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [43], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{43}', 'transitive_degree': 43, 'wreath_data': None, 'wreath_product': False}
