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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1152.32548', 'ambient_counter': 32548, 'ambient_order': 1152, 'ambient_tex': 'C_{12}^2:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 36, 'characteristic': True, 'core_order': 48, 'counter': 200, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1152.32548.24.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '24.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '24.14', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 14, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 24, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times D_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '48.45', 'subgroup_hash': 45, 'subgroup_order': 48, 'subgroup_tex': 'C_6\\times D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1152.32548', 'aut_centralizer_order': 144, 'aut_label': '24.c1', 'aut_quo_index': 12, 'aut_stab_index': 1, 'aut_weyl_group': '128.1755', 'aut_weyl_index': 144, 'centralizer': '32.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.c1.a1', '12.a1.a1', '12.d1.a1', '12.e1.a1', '12.o1.a1', '12.o1.b1', '12.q1.a1', '12.q1.b1'], 'contains': ['48.a1.a1', '48.a1.b1', '48.c1.a1', '48.bd1.a1', '72.a1.a1'], 'core': '24.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4165, 5819, 2491, 5851, 4230, 5467, 2389, 4186], 'generators': [4, 48, 600, 576, 384], 'label': '1152.32548.24.c1.a1', 'mobius_quo': 0, 'mobius_sub': 24, 'normal_closure': '24.c1.a1', 'normal_contained_in': ['8.c1.a1', '12.a1.a1', '12.d1.a1', '12.e1.a1'], 'normal_contains': ['48.a1.b1', '48.a1.a1', '48.c1.a1', '72.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '24.c1.a1', 'projective_image': '576.3412', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.c1.a1', 'subgroup_fusion': None, 'weyl_group': '32.27'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', '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, 2, 2, 2], 'aut_gens': [[1, 2, 4], [1, 2, 5], [25, 14, 44], [1, 38, 28], [1, 2, 20], [1, 27, 28], [1, 3, 4], [1, 26, 4]], 'aut_group': '128.1755', 'aut_hash': 1755, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 128, 'aut_permdeg': 10, 'aut_perms': [21025, 2810575, 374407, 1, 183120, 21288, 1270440], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [3, 1, 2, 1], [4, 2, 2, 1], [6, 1, 2, 1], [6, 1, 4, 1], [6, 2, 8, 1], [12, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.202', 'autcent_hash': 202, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3:D_4', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [3, 1, 2], [4, 2, 2], [6, 1, 6], [6, 2, 8], [12, 2, 4]], 'center_label': '12.5', 'center_order': 12, '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', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 45, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [3, 1, 2, 1], [4, 2, 1, 2], [6, 1, 2, 3], [6, 2, 2, 4], [12, 2, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 273, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '24.15', 'hash': 45, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2], 'inner_gens': [[1, 2, 4], [1, 2, 28], [1, 26, 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, 24], [2, 6]], 'label': '48.45', 'linC_count': 64, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 8, 'linQ_dim': 4, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6*D4', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 30, 'number_divisions': 20, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 54, 'number_subgroups': 70, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 11], [3, 2], [4, 4], [6, 22], [12, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 4, 0, 0], 'outer_gens': [[1, 2, 20], [25, 27, 20], [25, 39, 20], [25, 14, 45]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [7, 23, 5183, 15136], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 2]], 'representations': {'PC': {'code': 4858585473, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -3, 217, 42, 58]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931072, 16563488, 7233423]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [34649835, 11532102, 34603218, 34591355, 34649828]}, 'Perm': {'d': 9, 'gens': [41040, 10824, 90744, 4, 90720]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times D_4', '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': '8.5', '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, 6, 2, 24, 6, 24], 'aut_gens': [[1, 2, 8, 96], [29, 862, 1000, 672], [437, 390, 584, 720], [769, 166, 56, 120], [649, 1102, 424, 744], [773, 350, 200, 1056], [601, 1018, 56, 720], [29, 758, 584, 720]], 'aut_group': None, 'aut_hash': 7300551885917678560, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 18432, 'aut_permdeg': 72, 'aut_perms': [34786282839685982234641494697926838692852320959999007251867745726607623722858834724869115030720596316074, 56587190479693179451187230042465559032188776364314220044983164210780132889027042583049404041055328848404, 54955414242345715692932834724885608943211108037776521865621298819195114214478018799857349708422293512961, 45459199065906732492544149528738119145322343276664584726296972536788721384861479780217532354340308840322, 17747662554149414496335202604213149587161543462994284151529650912201070140166959588461133988097479706672, 38970458641066632677556717089898329343274110794386373208644235554530792709232703104641482211896631268086, 34303347958104518209298522614850954930041415981119042465114799492103239968267537692996434539893103627785], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 2, 1], [2, 8, 1, 1], [2, 12, 4, 1], [2, 24, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 4, 1, 1], [4, 4, 2, 1], [4, 24, 1, 1], [4, 24, 2, 1], [4, 144, 2, 1], [6, 2, 1, 2], [6, 2, 2, 1], [6, 4, 1, 2], [6, 4, 2, 1], [6, 8, 2, 3], [6, 12, 8, 1], [6, 16, 1, 1], [6, 16, 2, 2], [6, 24, 2, 1], [8, 48, 1, 1], [12, 4, 2, 1], [12, 4, 4, 1], [12, 8, 1, 1], [12, 8, 2, 2], [12, 8, 4, 1], [12, 24, 2, 1], [12, 24, 4, 1], [24, 48, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6^2.C_2^6.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '2304.nu', 'autcentquo_hash': 6703961864059093320, 'autcentquo_nilpotent': False, 'autcentquo_order': 2304, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^4:D_6^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 2], [2, 8, 1], [2, 12, 4], [2, 24, 1], [3, 2, 2], [3, 4, 1], [4, 4, 3], [4, 24, 3], [4, 144, 2], [6, 2, 4], [6, 4, 4], [6, 8, 6], [6, 12, 8], [6, 16, 5], [6, 24, 2], [8, 48, 1], [12, 4, 6], [12, 8, 9], [12, 24, 6], [24, 48, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '576.3412', 'commutator_count': 1, 'commutator_label': '144.179', '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', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 32548, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 2], [2, 8, 1, 1], [2, 12, 1, 4], [2, 24, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 4, 1, 3], [4, 24, 1, 3], [4, 144, 1, 2], [6, 2, 1, 2], [6, 2, 2, 1], [6, 4, 1, 2], [6, 4, 2, 1], [6, 8, 1, 4], [6, 8, 2, 1], [6, 12, 2, 4], [6, 16, 1, 3], [6, 16, 2, 1], [6, 24, 2, 1], [8, 48, 1, 1], [12, 4, 2, 3], [12, 8, 1, 3], [12, 8, 2, 3], [12, 24, 2, 3], [24, 48, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 21504, 'exponent': 24, 'exponents_of_order': [7, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 0, 4]], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '72.46', 'hash': 32548, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 6, 12], 'inner_gens': [[1, 6, 776, 1128], [5, 2, 664, 504], [385, 594, 8, 96], [217, 842, 8, 96]], 'inner_hash': 3412, 'inner_nilpotent': False, 'inner_order': 576, 'inner_split': True, 'inner_tex': '(C_6\\times C_{12}):D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 30], [4, 28], [8, 9]], 'label': '1152.32548', 'linC_count': 64, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 80, 'linQ_dim': 8, 'linQ_dim_count': 72, 'linR_count': 72, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C12^2:D4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 39, 'number_characteristic_subgroups': 42, 'number_conjugacy_classes': 75, 'number_divisions': 56, 'number_normal_subgroups': 68, 'number_subgroup_autclasses': 413, 'number_subgroup_classes': 751, 'number_subgroups': 5350, 'old_label': None, 'order': 1152, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 91], [3, 8], [4, 372], [6, 296], [8, 48], [12, 240], [24, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[5, 6, 248, 480], [49, 2, 472, 672], [49, 2, 472, 720], [649, 338, 776, 528]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [7, 23, 11536, 16583], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 20], [8, 11], [16, 3]], 'representations': {'PC': {'code': '535485172627763321441170547956824937627631362158187866004619831125835756553', 'gens': [1, 2, 4, 7], 'pres': [9, 2, 2, 2, 2, 2, 3, 2, 2, 3, 109, 46, 27939, 11964, 525, 102, 18004, 1813, 130, 43205, 1742, 71070, 15891, 10608, 186, 72583, 36304, 214, 62216, 31121]}, 'Perm': {'d': 14, 'gens': [6314117167, 13894761016, 1960645816, 11329920, 20757824640, 1480636080, 13902013440, 311, 56]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}^2:D_4', 'transitive_degree': 48, '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': '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, 3, 2, 2], 'aut_gens': [[1, 2, 4], [1, 2, 20], [1, 2, 5], [12, 15, 5], [1, 10, 4], [1, 14, 4], [1, 15, 4]], 'aut_group': '144.183', 'aut_hash': 183, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 144, 'aut_permdeg': 7, 'aut_perms': [1, 120, 144, 3, 744, 1680], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 4, 1], [3, 2, 1, 1], [6, 2, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', '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], [2, 3, 4], [3, 2, 1], [6, 2, 3]], 'center_label': '4.2', 'center_order': 4, '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', '2.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [3, 2, 1, 1], [6, 2, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 28, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '24.14', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 3], 'inner_gens': [[1, 2, 4], [1, 2, 20], [1, 10, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 4]], 'label': '24.14', 'linC_count': 12, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 12, 'linQ_dim': 3, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D6', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 12, 'number_divisions': 12, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 32, 'number_subgroups': 54, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 15], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 2, 5], [13, 14, 17], [1, 15, 4], [1, 14, 4]], 'outer_group': '24.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 4, 'outer_perms': [2, 4, 16, 7], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4]], 'representations': {'PC': {'code': 5123137, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -3, 126, 34, 135]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16325, 16295, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8156, 16849, 13286]}, 'Perm': {'d': 7, 'gens': [127, 7, 16, 840]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times D_6', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
