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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '960.2954', 'ambient_counter': 2954, 'ambient_order': 960, 'ambient_tex': '(C_{12}\\times D_{10}):C_4', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 12, 'counter': 247, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.2954.40.m1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '40.m1.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': 40, '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': '24.9', 'subgroup_hash': 9, 'subgroup_order': 24, 'subgroup_tex': 'C_2\\times C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.2954', 'aut_centralizer_order': None, 'aut_label': '40.m1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '10.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.o1.a1', '20.f1.a1', '20.h1.a1', '20.i1.a1'], 'contains': ['80.d1.a1', '80.i1.a1', '80.j1.a1', '120.m1.a1'], 'core': '80.d1.a1', 'coset_action_label': None, 'count': 10, 'diagramx': None, 'generators': [1, 240, 320, 480], 'label': '960.2954.40.m1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.f1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '10.b1.a1', 'old_label': '40.m1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '40.m1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '24.9', '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, 2], 'aut_gens': [[1, 2], [13, 2], [1, 23], [1, 10], [1, 14]], 'aut_group': '16.11', 'aut_hash': 11, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 6, 'aut_perms': [288, 6, 127, 126], '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, 4, 1], [6, 1, 2, 1], [6, 1, 4, 1], [12, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.11', 'autcent_hash': 11, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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], [12, 1, 8]], 'center_label': '24.9', 'center_order': 24, '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', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['4.1', 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], [12, 1, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '12.5', 'hash': 9, '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, 24]], 'label': '24.9', 'linC_count': 96, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 4, 'linQ_dim': 4, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C12', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 24, 'number_divisions': 12, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [12, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[13, 2], [1, 23], [1, 10], [1, 14]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [288, 6, 127, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 2]], 'representations': {'PC': {'code': 221281, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -3, 21, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931072, 26129103]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [10993, 4396]}, 'Perm': {'d': 9, 'gens': [2400, 40320, 4, 744]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], '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_{12}', 'transitive_degree': 24, '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, 10, 6, 4, 2, 4, 10, 2, 2, 6], 'aut_gens': [[1, 2, 4, 80], [553, 2, 86, 402], [497, 2, 46, 442], [41, 2, 646, 82], [25, 2, 598, 442], [545, 2, 356, 920], [491, 2, 854, 922], [539, 2, 244, 560], [545, 2, 198, 400], [483, 2, 404, 880], [523, 2, 398, 562]], 'aut_group': None, 'aut_hash': 352979904559946459, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 122880, 'aut_permdeg': 88, 'aut_perms': [166609098459525663583776391705517493078022005787867493587491805782948057851526601594033742717619046674014855982085802464049129235526935, 54150703715441451972728470354695437544367825920513899954029210455859024693891392787528667210191294098190245932837236872943163642810074, 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47271728876499221433490303297490501015140579975052917726071670491960262276647801084239079766670458438609250672936667643977192472099798], 'aut_phi_ratio': 480.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 10, 4, 1], [3, 2, 1, 1], [4, 2, 4, 1], [4, 10, 4, 1], [4, 12, 4, 1], [4, 60, 4, 1], [5, 2, 2, 1], [6, 2, 1, 7], [6, 10, 8, 1], [10, 2, 2, 7], [12, 2, 8, 1], [12, 10, 8, 1], [15, 4, 2, 1], [20, 4, 8, 1], [20, 12, 16, 1], [30, 4, 2, 7], [60, 4, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{15}:(C_2^8.C_2^5)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '512.10494213', 'autcent_hash': 1718285292446712970, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^9', '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, 7], [2, 10, 4], [3, 2, 1], [4, 2, 4], [4, 10, 4], [4, 12, 4], [4, 60, 4], [5, 2, 2], [6, 2, 7], [6, 10, 8], [10, 2, 14], [12, 2, 8], [12, 10, 8], [15, 4, 2], [20, 4, 8], [20, 12, 16], [30, 4, 14], [60, 4, 16]], 'center_label': '8.5', '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': 2954, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 10, 1, 4], [3, 2, 1, 1], [4, 2, 1, 4], [4, 10, 2, 2], [4, 12, 2, 2], [4, 60, 2, 2], [5, 2, 2, 1], [6, 2, 1, 7], [6, 10, 2, 4], [10, 2, 2, 7], [12, 2, 2, 4], [12, 10, 4, 2], [15, 4, 2, 1], [20, 4, 2, 4], [20, 12, 8, 2], [30, 4, 2, 7], [60, 4, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2016, 'exponent': 60, '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': '8.5', 'frattini_quotient': '120.42', 'hash': 2954, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 1, 10, 6], 'inner_gens': [[1, 2, 38, 80], [1, 2, 4, 80], [51, 2, 4, 880], [1, 2, 164, 80]], 'inner_hash': 42, 'inner_nilpotent': False, 'inner_order': 120, 'inner_split': False, 'inner_tex': 'S_3\\times D_{10}', 'inner_used': [1, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 76], [4, 40]], 'label': '960.2954', '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': '(C12*D10):C4', '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': 43, 'number_characteristic_subgroups': 78, 'number_conjugacy_classes': 132, 'number_divisions': 66, 'number_normal_subgroups': 160, 'number_subgroup_autclasses': 260, 'number_subgroup_classes': 468, 'number_subgroups': 2376, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 47], [3, 2], [4, 336], [5, 4], [6, 94], [10, 28], [12, 96], [15, 8], [20, 224], [30, 56], [60, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 4, 2, 2, 2, 4], 'outer_gen_pows': [320, 640, 320, 0, 16, 16, 48, 640], 'outer_gens': [[505, 2, 878, 600], [513, 2, 238, 560], [49, 2, 652, 560], [481, 2, 550, 400], [537, 2, 684, 440], [19, 2, 204, 442], [49, 2, 326, 920], [33, 2, 638, 600]], 'outer_group': None, 'outer_hash': 8224544021644802449, 'outer_nilpotent': True, 'outer_order': 1024, 'outer_permdeg': 256, 'outer_perms': 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235351865638911850447227805744527148268308502729748489397339327209518844543378187148118094466881827883227441406146171143301725877077588027775830272601520449641815643721619608020104141247318458960215726025155802982710377050145848478287894255766683704684767967629491955208189185853749006173593177199033938723866756911704410727962459197788570186168680346303214471188733674499400132504446620733100887103765436591294551750731782260109598162351014796134526914927704081151228419752720273333093275870633017062144656], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4.C_2^6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 14], [8, 20], [16, 4]], 'representations': {'PC': {'code': 1562705296696555529473398322988441787957441, 'gens': [1, 2, 3, 6], 'pres': [8, -2, -2, 2, -2, -5, -2, -2, -3, 914, 66, 2307, 91, 2564, 10581, 141, 11222, 166, 10263]}, 'GLZN': {'d': 2, 'p': 60, 'gens': [216721, 10695271, 10584049, 3006023, 8929201, 6858916, 8856041, 4104019]}, 'Perm': {'d': 20, 'gens': [21009968184365, 6446922295161609, 961632016, 21009968179216, 16, 1520467200, 128047474114560000, 50520]}}, 'schur_multiplier': [2, 2, 2, 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_{12}\\times D_{10}):C_4', 'transitive_degree': 480, 'wreath_data': None, 'wreath_product': False}
