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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '145200.l', 'ambient_counter': 12, 'ambient_order': 145200, 'ambient_tex': 'C_{220}:F_{11}:S_3', 'central': False, 'central_factor': False, 'centralizer_order': 100, 'characteristic': False, 'core_order': 2, 'counter': 289, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '145200.l.12100.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12100.a1', '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': 12100, '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': '145200.l', 'aut_centralizer_order': None, 'aut_label': '12100.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1452.b1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['100.a1', '2420.a1', '2420.b1', '6050.a1', '6050.b1', '6050.c1'], 'contains': ['24200.a1', '24200.b1', '36300.b1'], 'core': '72600.a1', 'coset_action_label': None, 'count': 484, 'diagramx': [4203, -1, 3534, -1], 'generators': [54155, 72600, 80900], 'label': '145200.l.12100.a1', 'mobius_quo': None, 'mobius_sub': -10, 'normal_closure': '50.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '121.a1', 'old_label': '12100.a1', 'projective_image': '72600.s', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12100.a1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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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': '200.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [20, 20, 60, 60, 20, 60, 10], 'aut_gens': [[1, 10, 300, 3300], [125683, 61010, 93000, 29700], [44615, 121692, 81600, 10500], [39329, 141438, 80100, 30300], [136097, 127156, 120600, 123000], [30247, 66556, 1500, 16500], [51597, 106156, 27600, 131100], [110733, 82492, 26700, 143700]], 'aut_group': None, 'aut_hash': 5227947440930426124, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4646400, 'aut_permdeg': 869, 'aut_perms': 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'aut_phi_ratio': 132.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 66, 4, 1], [2, 121, 2, 1], [3, 242, 1, 1], [4, 2, 1, 1], [4, 242, 1, 1], [5, 1, 4, 1], [5, 121, 5, 4], [6, 242, 1, 1], [6, 242, 2, 1], [10, 1, 4, 1], [10, 66, 16, 1], [10, 121, 5, 4], [10, 121, 8, 1], [10, 121, 10, 4], [10, 726, 20, 4], [11, 30, 2, 1], [11, 60, 1, 1], [12, 242, 2, 2], [15, 242, 4, 1], [15, 242, 5, 4], [20, 2, 4, 1], [20, 242, 4, 1], [20, 242, 5, 8], [22, 30, 2, 1], [22, 60, 1, 1], [22, 660, 4, 1], [30, 242, 4, 1], [30, 242, 5, 4], [30, 242, 8, 1], [30, 242, 10, 4], [44, 60, 2, 2], [55, 30, 8, 1], [55, 60, 4, 1], [60, 242, 8, 2], [60, 242, 10, 8], [110, 30, 8, 1], [110, 60, 4, 1], [110, 660, 16, 1], [220, 60, 8, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_{11}^2.C_{30}.C_{10}.C_2^6.C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '160.236', 'autcent_hash': 236, 'autcent_nilpotent': False, 'autcent_order': 160, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times F_5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 660, 'autcentquo_group': None, 'autcentquo_hash': 9201191957465338480, 'autcentquo_nilpotent': False, 'autcentquo_order': 29040, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{11}^2.C_{30}.C_2^3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 66, 4], [2, 121, 2], [3, 242, 1], [4, 2, 1], [4, 242, 1], [5, 1, 4], [5, 121, 20], [6, 242, 3], [10, 1, 4], [10, 66, 16], [10, 121, 68], [10, 726, 80], [11, 30, 2], [11, 60, 1], [12, 242, 4], [15, 242, 24], [20, 2, 4], [20, 242, 44], [22, 30, 2], [22, 60, 1], [22, 660, 4], [30, 242, 72], [44, 60, 4], [55, 30, 8], [55, 60, 4], [60, 242, 96], [110, 30, 8], [110, 60, 4], [110, 660, 16], [220, 60, 16]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '14520.bk', 'commutator_count': 1, 'commutator_label': '726.7', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '5.1', '5.1', '11.1', '11.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['29040.v', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 66, 1, 4], [2, 121, 1, 2], [3, 242, 1, 1], [4, 2, 1, 1], [4, 242, 1, 1], [5, 1, 4, 1], [5, 121, 4, 5], [6, 242, 1, 3], [10, 1, 4, 1], [10, 66, 4, 4], [10, 121, 4, 17], [10, 726, 4, 20], [11, 30, 1, 2], [11, 60, 1, 1], [12, 242, 2, 2], [15, 242, 4, 6], [20, 2, 4, 1], [20, 242, 4, 11], [22, 30, 1, 2], [22, 60, 1, 1], [22, 660, 1, 4], [30, 242, 4, 18], [44, 60, 1, 2], [44, 60, 2, 1], [55, 30, 4, 2], [55, 60, 4, 1], [60, 242, 8, 12], [110, 30, 4, 2], [110, 60, 4, 1], [110, 660, 4, 4], [220, 60, 4, 2], [220, 60, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 182988288, 'exponent': 660, 'exponents_of_order': [4, 2, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[60, 0, 16]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '72600.s', 'hash': 5524277664530726842, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [10, 30, 11, 22], 'inner_gens': [[1, 81710, 120900, 51600], [40401, 10, 133500, 124200], [27901, 15310, 300, 3300], [100201, 27610, 300, 3300]], 'inner_hash': 3901236506553590707, 'inner_nilpotent': False, 'inner_order': 14520, 'inner_split': False, 'inner_tex': 'C_{11}^2:(C_{10}\\times D_6)', 'inner_used': [1, 2, 4], 'irrC_degree': 60, 'irrQ_degree': 240, 'irrQ_dim': 240, 'irrR_degree': None, 'irrep_stats': [[1, 200], [2, 250], [30, 40], [60, 30]], 'label': '145200.l', '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': 'C220:F11:S3', 'ngens': 9, '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, 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, 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': 80, 'number_characteristic_subgroups': 45, 'number_conjugacy_classes': 520, 'number_divisions': 138, 'number_normal_subgroups': 222, 'number_subgroup_autclasses': 320, 'number_subgroup_classes': 1360, 'number_subgroups': 132208, 'old_label': None, 'order': 145200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 507], [3, 242], [4, 244], [5, 2424], [6, 726], [10, 67368], [11, 120], [12, 968], [15, 5808], [20, 10656], [22, 2760], [30, 17424], [44, 240], [55, 480], [60, 23232], [110, 11040], [220, 960]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 20, 'outer_gen_orders': [2, 2, 2, 4, 10], 'outer_gen_pows': [92520, 0, 92520, 0, 0], 'outer_gens': [[14101, 28010, 27600, 96900], [120301, 114010, 3000, 69300], [105751, 107210, 15000, 17400], [42065, 134892, 3000, 69300], [92015, 88974, 106800, 96300]], 'outer_group': '320.1638', 'outer_hash': 1638, 'outer_nilpotent': False, 'outer_order': 320, 'outer_permdeg': 13, 'outer_perms': [5167, 11536, 16, 43913640, 1444625303], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times F_5', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 42, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 50], [8, 36], [16, 12], [30, 8], [60, 4], [120, 9], [240, 4], [480, 1]], 'representations': {'PC': {'code': '10172080363948937544176809577114425343036235643051253766166816746862688859272058678959419115053709749632695725614451783512940150345843001215447912414805695066686806294221030139009', 'gens': [1, 3, 6, 7], 'pres': [9, 2, 5, 2, 3, 5, 11, 2, 2, 11, 18, 2206172, 214796, 74, 51123, 1690572, 138, 1285204, 2093863, 6528605, 32414, 720923, 77792, 2471, 3250806, 3846165, 782484, 225888, 31227, 186, 7192807, 3564016, 719305, 504394, 71323, 214, 4422608, 2138417, 442286, 546785, 160424]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 30551766872799773, 16267516240797404, 44332910602637236, 10522401710269773, 5063783017457019, 3352440040187918, 33418192856010432, 4190593668098319]}, 'Perm': {'d': 42, 'gens': [33, 4269122717314680267192180673103808913186964090944, 90753, 38435947295107606330666132746755107316813124230464, 72786475703563735937732166174855939129277413619290, 101381843496471307778764776351008377898253870393664, 137139285381337746165596084963912371070587739361216, 6697983241494594593294770090912237850027561011200, 174838111355488660326455349895823905948142837493840]}}, 'schur_multiplier': [2, 2, 10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10, 10], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{220}:F_{11}:S_3', 'transitive_degree': 660, 'wreath_data': None, 'wreath_product': False}
