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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.53634', 'ambient_counter': 53634, 'ambient_order': 256, 'ambient_tex': 'C_4^2.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 8, 'counter': 126, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '256.53634.16.bh1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '16.bh1', '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': 16, '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': '16.12', 'subgroup_hash': 12, 'subgroup_order': 16, 'subgroup_tex': 'C_2\\times Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.53634', 'aut_centralizer_order': 256, 'aut_label': '16.bh1', 'aut_quo_index': None, 'aut_stab_index': 12, 'aut_weyl_group': '64.138', 'aut_weyl_index': 3072, 'centralizer': '16.v1', 'complements': None, 'conjugacy_class_count': 6, 'contained_in': ['8.f1', '8.bk1', '8.bp1', '8.bq1', '8.bs1'], 'contains': ['32.c1', '32.r1', '32.bc1'], 'core': '32.c1', 'coset_action_label': None, 'count': 12, 'diagramx': [3830, -1, 3847, -1], 'generators': [65, 98, 16], 'label': '256.53634.16.bh1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.f1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.h1', 'old_label': '16.bh1', 'projective_image': '64.261', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.bh1', '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': '8.5', '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, 3, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [1, 10, 6], [9, 2, 4], [1, 6, 10], [1, 11, 13], [1, 11, 4], [1, 10, 4], [1, 2, 12]], 'aut_group': '192.955', 'aut_hash': 955, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 8, 'aut_perms': [55, 12316, 3018, 27630, 18246, 18498, 5329], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr 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, 3], [4, 2, 6]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 7, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 12, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2], 'inner_gens': [[1, 2, 4], [1, 2, 12], [1, 10, 4]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.12', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 8, 'linQ_dim': 5, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*Q8', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 10, 'number_divisions': 10, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 19, 'number_subgroups': 19, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [2, 0, 0, 0, 0], 'outer_gens': [[1, 2, 6], [9, 2, 4], [1, 14, 2], [1, 3, 5], [1, 3, 4]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [403, 316, 181, 314, 82], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2]], 'representations': {'PC': {'code': 139846, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 37, 78, 34]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [706188457522, 141602720900, 706104070413]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [16990, 16120, 13286, 13884]}, 'Perm': {'d': 10, 'gens': [465966, 859974, 1, 1275486]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times Q_8', 'transitive_degree': 16, '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [4, 6, 4, 12, 4, 12], 'aut_gens': [[1, 2, 4, 8, 32], [145, 10, 132, 136, 104], [197, 219, 17, 24, 112], [20, 66, 145, 216, 120], [132, 214, 197, 136, 120], [129, 18, 148, 200, 248], [4, 22, 85, 24, 176]], 'aut_group': None, 'aut_hash': 1917651590489209160, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 196608, 'aut_permdeg': 44, 'aut_perms': [2267409124871128518173752332018082737746624912365845016, 505127094420190134735940596396551934084539996055742407, 1078782989399918063770434880723041753809839822485348442, 1341620668323546286154739401148584729260461912891971328, 1321472958483013874501769616914964123948654660328676111, 67394867802306745323151336828581828606094613120893939], 'aut_phi_ratio': 1536.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 6, 1], [2, 8, 2, 1], [2, 8, 6, 1], [4, 2, 2, 1], [4, 2, 4, 1], [4, 2, 6, 1], [4, 4, 6, 1], [4, 8, 2, 1], [4, 8, 6, 1], [8, 2, 8, 1], [8, 4, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5.(C_2^5\\times C_6).C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '1024.djt', 'autcent_hash': 1957374279625110839, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '192.1147', 'autcentquo_hash': 1147, 'autcentquo_nilpotent': False, 'autcentquo_order': 192, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^4:D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 6], [2, 8, 8], [4, 2, 12], [4, 4, 6], [4, 8, 8], [8, 2, 8], [8, 4, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '64.261', 'commutator_count': 1, 'commutator_label': '8.2', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 53634, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 6], [2, 8, 1, 8], [4, 2, 1, 12], [4, 4, 1, 6], [4, 8, 1, 8], [8, 2, 2, 4], [8, 4, 1, 12]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1666560, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '32.51', 'hash': 53634, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 2, 4], 'inner_gens': [[1, 2, 132, 8, 32], [1, 2, 4, 24, 96], [129, 2, 4, 8, 32], [1, 18, 4, 8, 32], [1, 194, 4, 8, 32]], 'inner_hash': 261, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': False, 'inner_tex': 'D_4\\times C_2^3', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [4, 8]], 'label': '256.53634', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.C2^4', 'ngens': 5, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 64, 'number_divisions': 60, 'number_normal_subgroups': 563, 'number_subgroup_autclasses': 191, 'number_subgroup_classes': 1625, 'number_subgroups': 3535, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 79], [4, 112], [8, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 6, 4, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 72, 0, 0, 0, 0, 0, 7, 0, 0, 68], 'outer_gens': [[1, 26, 4, 24, 56], [17, 138, 20, 136, 160], [1, 2, 20, 24, 40], [1, 2, 4, 24, 40], [20, 198, 213, 24, 40], [1, 2, 20, 136, 168], [1, 2, 4, 8, 176], [17, 2, 20, 88, 40], [1, 2, 4, 24, 56], [17, 2, 20, 24, 56], [197, 198, 4, 8, 32]], 'outer_group': '3072.bhj', 'outer_hash': 6634943677507123677, 'outer_nilpotent': False, 'outer_order': 3072, 'outer_permdeg': 14, 'outer_perms': [6358746322, 12936672264, 45719699765, 45719700024, 45719700345, 7795837034, 27909186480, 39492674236, 32148633144, 32148633196, 134], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:D_4\\times S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 20, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 24], [8, 4]], 'representations': {'PC': {'code': 9748145590994084313858408, 'gens': [1, 2, 3, 4, 6], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 3170, 395, 91, 2317, 141, 5390, 166]}, 'Perm': {'d': 20, 'gens': [124178442194908205, 276169695258574200, 406280772296517009, 509009925758710440, 123163659110958600, 406280772296517000, 656816117483554576, 656816117483554560]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.C_2^4', 'transitive_degree': 64, 'wreath_data': None, 'wreath_product': False}
