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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '768.323569', 'ambient_counter': 323569, 'ambient_order': 768, 'ambient_tex': 'D_4^2:D_6', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 4, 'counter': 351, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '768.323569.48.bl1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '48.bl1', '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': 48, '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': '16.10', 'subgroup_hash': 10, 'subgroup_order': 16, 'subgroup_tex': 'C_2^2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '768.323569', 'aut_centralizer_order': None, 'aut_label': '48.bl1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '8.o1', 'complements': None, 'conjugacy_class_count': 8, 'contained_in': ['16.bl1', '24.l1', '24.bf1', '24.bg1', '24.bm1', '24.bn1', '24.bo1', '24.bp1'], 'contains': ['96.q1', '96.t1', '96.u1', '96.cf1'], 'core': '192.c1', 'coset_action_label': None, 'count': 16, 'diagramx': None, 'generators': [1078852320, 164516401, 1078127281, 123752880], 'label': '768.323569.48.bl1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '12.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.c1', 'old_label': '48.bl1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '48.bl1', '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [10, 9, 7], [10, 3, 15], [1, 2, 5], [1, 2, 14], [9, 2, 14], [1, 10, 13], [1, 2, 12]], 'aut_group': '192.1493', 'aut_hash': 1493, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 8, 'aut_perms': [17764, 31287, 5329, 40319, 37965, 21769, 12316], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [4, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '192.1493', 'autcent_hash': 1493, 'autcent_nilpotent': False, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3: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]], 'center_label': '16.10', 'center_order': 16, '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'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 1, 2, 4]], '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': 10, '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, 16]], 'label': '16.10', 'linC_count': 224, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 48, 'linQ_dim': 4, 'linQ_dim_count': 48, 'linR_count': 48, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C4', 'ngens': 3, 'nilpotency_class': 1, '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': 16, 'number_divisions': 12, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 27, 'number_subgroups': 27, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[10, 9, 7], [10, 3, 15], [1, 2, 5], [1, 2, 14], [9, 2, 14], [1, 10, 13], [1, 2, 12]], 'outer_group': '192.1493', 'outer_hash': 1493, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 8, 'outer_perms': [17764, 31287, 5329, 40319, 37965, 21769, 12316], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4]], 'representations': {'PC': {'code': 516, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7115326, 7115160]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [1173753, 1832002, 438254, 1565004]}, 'Perm': {'d': 8, 'gens': [22, 5040, 120, 7]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], '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_4', '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [4, 4, 12, 6, 2, 12, 8, 12], 'aut_gens': [[486985687, 1078127281, 163780561, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 30], [2614308505, 124477921, 1199087280, 1078852321, 997930800, 3085331760, 123752880, 1078852320, 48], [2000195287, 1134001, 1199087280, 1, 1139040, 3085331760, 123752880, 1078852320, 30], [2000931144, 1571280480, 1078847280, 1078852321, 3125253600, 1139040, 123752880, 1078852320, 30], [2533387014, 124477921, 1118880001, 1, 997930800, 3125253600, 123752880, 1078852320, 30], [2000931127, 124477920, 163780560, 1078852321, 1078116480, 1611308160, 123752880, 1078852320, 30], [2000931175, 1134000, 243578881, 1, 1139040, 3125253600, 123752880, 1078852320, 30], [487721527, 1491482161, 1078847280, 1078852321, 1694781360, 1139040, 123752880, 1078852320, 48], [2000195335, 10801, 3045339360, 1, 1139040, 1694781360, 999054000, 1078852320, 30]], 'aut_group': None, 'aut_hash': 1175629290017836394, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 49152, 'aut_permdeg': 80, 'aut_perms': [62493108848273537796408119833003035345992991925526070777436585952435883291390791226725364057085296686889542509396081652, 49743180898940446484652068346080293312275382693143719126812481568376991675968594396510933511325557029847282015834928122, 54565385121341164751998802056309546179099775221914065273825848444742671010154908780788902448895989619705462648877349360, 44305386274441366362235091623443928475627264022458088342221043787984725706770964788938381467543560509562397388447920172, 22382210920532394527056320902568866391247634584321995551321153830827366611885135508661783181758025850435424691511651722, 57830873232311408779434124770033137107122715404779386438152926502795137675354929691258607966493570840834491376972150319, 4762374219428066927982636882909123548736339632809015225294900881211207909426431580706345519314318120593115634912059386, 19581270759944520766690743591125236750903869819835186784025706212395394903234949722086338946310366079235435160479712479], 'aut_phi_ratio': 192.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 1, 2], [2, 4, 2, 2], [2, 4, 8, 1], [2, 8, 2, 1], [2, 24, 2, 1], [3, 2, 1, 1], [4, 4, 1, 2], [4, 4, 4, 1], [4, 8, 4, 1], [4, 24, 2, 1], [4, 48, 4, 1], [6, 2, 1, 1], [6, 2, 2, 3], [6, 4, 16, 1], [6, 8, 2, 2], [6, 8, 4, 1], [8, 48, 2, 1], [12, 4, 8, 1], [12, 8, 1, 2], [12, 8, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^6.C_2^6.C_2^2)', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '384.12882', 'autcentquo_hash': 12882, 'autcentquo_nilpotent': False, 'autcentquo_order': 384, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\wr C_2^2\\times S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 12], [2, 8, 2], [2, 24, 2], [3, 2, 1], [4, 4, 6], [4, 8, 4], [4, 24, 2], [4, 48, 4], [6, 2, 7], [6, 4, 16], [6, 8, 8], [8, 48, 2], [12, 4, 8], [12, 8, 10]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '192.803', 'commutator_count': 1, 'commutator_label': '48.45', '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'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 323569, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['384.4485', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 12], [2, 8, 1, 2], [2, 24, 1, 2], [3, 2, 1, 1], [4, 4, 1, 6], [4, 8, 1, 4], [4, 24, 1, 2], [4, 48, 1, 4], [6, 2, 1, 3], [6, 2, 2, 2], [6, 4, 2, 8], [6, 8, 1, 4], [6, 8, 2, 2], [8, 48, 1, 2], [12, 4, 2, 4], [12, 8, 1, 2], [12, 8, 2, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2096640, 'exponent': 24, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '48.51', 'hash': 323569, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 2, 1, 2, 2, 2, 1, 3], 'inner_gens': [[486985687, 1078847281, 1491482161, 1078852321, 1139040, 1611308160, 999054000, 1078852320, 48], [487721527, 1078127281, 163780561, 1078852321, 1139040, 3085331760, 123752880, 1078852320, 30], [2533386967, 1078127281, 163780561, 1078852321, 124472880, 1611308160, 123752880, 1078852320, 30], [486985687, 1078127281, 163780561, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 30], [486985687, 1078127281, 243578881, 1078852321, 1139040, 3125253600, 123752880, 1078852320, 30], [486985687, 124477921, 163780561, 1078852321, 1078116480, 1611308160, 123752880, 1078852320, 30], [2000931127, 1078127281, 163780561, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 30], [486985687, 1078127281, 163780561, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 30], [486985735, 1078127281, 163780561, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 30]], 'inner_hash': 803, 'inner_nilpotent': False, 'inner_order': 192, 'inner_split': False, 'inner_tex': '(C_2\\times C_{12}):D_4', 'inner_used': [1, 2, 3, 9], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 44], [4, 28], [8, 2]], 'label': '768.323569', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': True, 'name': 'D4^2:D6', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 37, 'number_conjugacy_classes': 90, 'number_divisions': 70, 'number_normal_subgroups': 171, 'number_subgroup_autclasses': 542, 'number_subgroup_classes': 1956, 'number_subgroups': 8964, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 119], [3, 2], [4, 296], [6, 142], [8, 96], [12, 112]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 123752880, 0, 0, 0, 0, 0, 0], 'outer_gens': [[2000931126, 1078127281, 1199087280, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 48], [2000931126, 1134001, 1199087281, 1, 1139040, 1611308160, 123752880, 1078852320, 30], [486985687, 243578881, 124477921, 1, 3125253600, 1078116480, 999054000, 1078852320, 30], [2000931126, 1078127281, 163780561, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 30], [486985687, 1078127281, 163780561, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 48], [486985687, 1078127281, 1199087281, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 30], [486985687, 1078127280, 1199087280, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 30], [486985687, 1134001, 1199087281, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 30]], 'outer_group': '256.53380', 'outer_hash': 53380, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 12, 'outer_perms': [87091200, 363000, 176158200, 7, 40284856, 40279680, 87107880, 40284840], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5:D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 24], [8, 10]], 'representations': {'PC': {'code': '1169604327960086432296589715123037678048943960958019585', 'gens': [1, 2, 3, 5, 7], 'pres': [9, 2, 2, 2, 2, 2, 2, 2, 2, 3, 326, 74, 4882, 301, 130, 45366, 5820, 3561, 186, 48391, 12121, 214, 41480, 10394]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101750001908316, 91793132065999401, 24992906222183252, 58415899998857945]}, 'Perm': {'d': 13, 'gens': [486985687, 1078127281, 163780561, 1078852321, 1139040, 1611308160, 123752880, 1078852320, 30]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4^2:D_6', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
