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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '480.1050', 'ambient_counter': 1050, 'ambient_order': 480, 'ambient_tex': 'C_{30}:C_4^2', 'central': False, 'central_factor': False, 'centralizer_order': 120, 'characteristic': True, 'core_order': 5, 'counter': 79, 'cyclic': True, 'direct': False, 'hall': 5, 'label': '480.1050.96.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '96.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '96.161', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 161, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 96, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_4\\times C_{12}', 'simple': True, 'solvable': True, 'special_labels': ['D', 'L1', 'D1', 'C6'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '5.1', 'subgroup_hash': 1, 'subgroup_order': 5, 'subgroup_tex': 'C_5', 'supersolvable': True, 'sylow': 5}
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gps_subgroup_data • Show schema
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{'ambient': '480.1050', 'aut_centralizer_order': 640, 'aut_label': '96.a1', 'aut_quo_index': 24, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 640, 'centralizer': '4.b1', 'complements': ['5.a1'], 'conjugacy_class_count': 1, 'contained_in': ['32.a1', '48.a1', '48.b1', '48.c1', '48.d1'], 'contains': ['480.a1'], 'core': '96.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2813, 6927, 3660, 4561], 'generators': [96], 'label': '480.1050.96.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '96.a1', 'normal_contained_in': ['32.a1', '48.a1', '48.b1', '48.c1', '48.d1'], 'normal_contains': ['480.a1'], 'normalizer': '1.a1', 'old_label': '96.a1', 'projective_image': '480.1050', 'quotient_action_image': '4.1', 'quotient_action_kernel': '24.9', 'quotient_action_kernel_order': 24, 'quotient_fusion': None, 'short_label': '96.a1', 'subgroup_fusion': None, 'weyl_group': '4.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '5.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [2]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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], [5, 1, 4]], 'center_label': '5.1', 'center_order': 5, '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': ['5.1'], '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], [5, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '5.1', 'hash': 1, 'hyperelementary': 5, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 5]], 'label': '5.1', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5', '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': 5, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 5, 'order_factorization_type': 1, 'order_stats': [[1, 1], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 5, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -5]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26329372]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131]}, 'Perm': {'d': 5, 'gens': [96]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [5], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '96.161', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [2, 2, 4, 2, 20, 2, 2, 2, 4, 2], 'aut_gens': [[1, 4, 16], [3, 245, 16], [3, 7, 176], [1, 244, 18], [1, 4, 176], [241, 52, 18], [1, 244, 176], [1, 6, 304], [243, 4, 16], [1, 5, 208], [1, 245, 18]], 'aut_group': '2560.dp', 'aut_hash': 7799020297190345341, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2560, 'aut_permdeg': 42, 'aut_perms': [104519257417329503780852476535802935605917772455779, 328658726735952224597354582071257148523671096929989, 565367574627706526543398238953678606197308090971113, 24942610584339699213767540736000000, 1220988835966186251473721628425058188185334335541366, 137175233975269412857488799856002729522259132677113, 1231908629846409822509559973823315687146625102609100, 536791836184784533309086645398362397639677551121730, 905493754506841223325989935560229950424024367062300, 130628120097498948787395207951598293946312352131637], 'aut_phi_ratio': 20.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 5, 2, 2], [3, 1, 2, 1], [4, 1, 4, 1], [4, 5, 4, 1], [4, 5, 8, 2], [5, 4, 1, 1], [6, 1, 2, 1], [6, 1, 4, 1], [6, 5, 4, 2], [10, 4, 1, 1], [10, 4, 2, 1], [12, 1, 8, 1], [12, 5, 8, 1], [12, 5, 16, 2], [15, 4, 2, 1], [20, 4, 4, 1], [30, 4, 2, 1], [30, 4, 4, 1], [60, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_4\\times F_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1755', 'autcent_hash': 1755, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4:D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 5, 4], [3, 1, 2], [4, 1, 4], [4, 5, 20], [5, 4, 1], [6, 1, 6], [6, 5, 8], [10, 4, 3], [12, 1, 8], [12, 5, 40], [15, 4, 2], [20, 4, 4], [30, 4, 6], [60, 4, 8]], 'center_label': '24.9', 'center_order': 24, 'central_product': True, 'central_quotient': '20.3', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1050, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['20.3', 1], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 5, 1, 4], [3, 1, 2, 1], [4, 1, 2, 2], [4, 5, 2, 10], [5, 4, 1, 1], [6, 1, 2, 3], [6, 5, 2, 4], [10, 4, 1, 3], [12, 1, 4, 2], [12, 5, 4, 10], [15, 4, 2, 1], [20, 4, 2, 2], [30, 4, 2, 3], [60, 4, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 13104, 'exponent': 60, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '240.200', 'hash': 1050, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [1, 4, 5], 'inner_gens': [[1, 4, 16], [1, 4, 112], [1, 388, 16]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'F_5', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 96], [4, 24]], 'label': '480.1050', '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': 'C30:C4^2', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 26, 'number_characteristic_subgroups': 28, 'number_conjugacy_classes': 120, 'number_divisions': 52, 'number_normal_subgroups': 124, 'number_subgroup_autclasses': 92, 'number_subgroup_classes': 216, 'number_subgroups': 584, 'old_label': None, 'order': 480, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 23], [3, 2], [4, 104], [5, 4], [6, 46], [10, 12], [12, 208], [15, 8], [20, 16], [30, 24], [60, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[243, 246, 176], [3, 7, 16], [3, 7, 18], [1, 4, 176], [1, 246, 16], [3, 246, 16], [1, 6, 16]], 'outer_group': '128.1755', 'outer_hash': 1755, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 10, 'outer_perms': [848167, 126774, 1356175, 1, 2328, 375894, 848928], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 16], [8, 6], [16, 2]], 'representations': {'PC': {'code': 8929468035917502759058582025628631047, 'gens': [1, 3, 5], 'pres': [7, -2, -2, -2, -2, -2, -3, -5, 14, 58, 998, 1355, 102, 2371, 698, 166, 2372, 2379]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [106591356994, 99673645696, 115652466179, 119218567616, 101195277308, 122109387504, 110232582984]}, 'Perm': {'d': 14, 'gens': [46089, 3719520, 3749760, 6706022400, 90736, 90720, 34]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{30}:C_4^2', 'transitive_degree': 120, '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': '96.161', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 2, 2, 6, 2, 2, 2, 4, 2], 'aut_gens': [[1, 2, 8], [49, 54, 56], [1, 50, 8], [1, 2, 40], [1, 75, 34], [1, 54, 8], [1, 54, 60], [1, 50, 56], [1, 26, 40], [53, 50, 12]], 'aut_group': '3072.bjj', 'aut_hash': 3278739220350440602, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3072, 'aut_permdeg': 26, 'aut_perms': [156599506467661408882560, 156290489358596340678150, 1, 267810879463645168873558446, 156598898240815284791046, 2508820767086692523195520, 49067179719652537489161384, 15618248412000798990155431, 49067179719799053264666840], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 1, 4, 1], [3, 1, 2, 1], [4, 1, 24, 1], [6, 1, 6, 1], [6, 1, 8, 1], [12, 1, 48, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^7:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '3072.bjj', 'autcent_hash': 3278739220350440602, 'autcent_nilpotent': False, 'autcent_order': 3072, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^7: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], [3, 1, 2], [4, 1, 24], [6, 1, 14], [12, 1, 48]], 'center_label': '96.161', 'center_order': 96, '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', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 161, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['4.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 1], [4, 1, 2, 12], [6, 1, 2, 7], [12, 1, 4, 12]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 91, 'exponent': 12, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '24.15', 'hash': 161, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 8], [1, 2, 8], [1, 2, 8]], '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, 96]], 'label': '96.161', 'linC_count': 46592, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 192, 'linQ_dim': 6, 'linQ_dim_count': 192, 'linR_count': 1536, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*C4*C12', '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 96, 'number_divisions': 40, 'number_normal_subgroups': 108, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 108, 'number_subgroups': 108, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 24], [6, 14], [12, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 2, 2, 6, 2, 2, 2, 4, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[49, 54, 56], [1, 50, 8], [1, 2, 40], [1, 75, 34], [1, 54, 8], [1, 54, 60], [1, 50, 56], [1, 26, 40], [53, 50, 12]], 'outer_group': '3072.bjj', 'outer_hash': 3278739220350440602, 'outer_nilpotent': False, 'outer_order': 3072, 'outer_permdeg': 26, 'outer_perms': [156599506467661408882560, 156290489358596340678150, 1, 267810879463645168873558446, 156598898240815284791046, 2508820767086692523195520, 49067179719652537489161384, 15618248412000798990155431, 49067179719799053264666840], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^7:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 12]], 'representations': {'PC': {'code': 638607951489, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -2, -2, -2, -3, 31, 69, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [91655872443238928, 125101750005088212, 58438776791332585]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [10319436418, 9789111396, 6526074264, 6878714910, 4548416880, 6762482606]}, 'Perm': {'d': 13, 'gens': [11612160, 2400, 479001600, 4, 3669120, 744]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 12], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4\\times C_{12}', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
