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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1728.31928', 'ambient_counter': 31928, 'ambient_order': 1728, 'ambient_tex': 'C_3\\times C_6.\\GL(2,\\mathbb{Z}/4)', 'central': True, 'central_factor': False, 'centralizer_order': 1728, 'characteristic': True, 'core_order': 1, 'counter': 374, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '1728.31928.1728.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '1728.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '1728.31928', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 31928, 'quotient_metabelian': False, 'quotient_nilpotent': False, 'quotient_order': 1728, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'C_3\\times C_6.\\GL(2,\\mathbb{Z}/4)', 'simple': False, 'solvable': True, 'special_labels': ['PC', 'U2', 'D3', 'C8'], 'split': True, 'standard_generators': False, 'stem': True, 'subgroup': '1.1', 'subgroup_hash': 1, 'subgroup_order': 1, 'subgroup_tex': 'C_1', 'supersolvable': True, 'sylow': 1}
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gps_subgroup_data • Show schema
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{'ambient': '1728.31928', 'aut_centralizer_order': None, 'aut_label': '1728.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1.a1', 'complements': ['1.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['576.a1.a1', '576.b1.a1', '576.c1.a1', '576.d1.a1', '576.e1.a1', '576.f1.a1', '576.g1.a1', '864.a1.a1', '864.b1.a1', '864.c1.a1'], 'contains': [], 'core': '1728.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3118, 6731, 2881, 5202, 1860, 6127, 3396, 6127], 'generators': [], 'label': '1728.31928.1728.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '1728.a1.a1', 'normal_contained_in': ['432.b1.a1', '576.a1.a1', '576.b1.a1', '864.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '1728.a1.a1', 'projective_image': '1728.31928', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1728.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', '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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 2, 3, 4, 2, 12, 2, 3, 12, 2, 2], 'aut_gens': [[3200423, 43332947, 742666, 40007587, 25487491], [27947621, 18883361, 25932628, 46827223, 890233], [3200423, 15055523, 742666, 40007587, 25492195], [3054011, 43779239, 742666, 39714763, 890233], [1546550, 27799226, 13189429, 22229965, 25492195], [28094033, 18439421, 1042546, 40007587, 25487491], [49517021, 4382129, 38379391, 47120047, 890233], [28094033, 18439421, 25639804, 40007587, 25487491], [3200423, 43337651, 742666, 40007587, 25487491], [26440160, 17125916, 38086567, 15117505, 25487491], [3200423, 43332947, 13489309, 47120047, 890233], [3200423, 43332947, 13489309, 47120047, 894937]], 'aut_group': '4608.lj', 'aut_hash': 3301011005301566777, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4608, 'aut_permdeg': 38, 'aut_perms': [355156898985373875270217154944527887631083916, 9178808122487176061880041086544783666064, 395719148671277575949775647747933550542986122, 507384920426145018326208072519368913835020973, 510013489425694822888677562393233702577716633, 255392521522775702417405120873680163374102578, 510013488364665702150430109762316140687951397, 5688177853474116378478890976342955512344, 467173995632557618329197725699906544355746150, 1587319532413219498617016401107210772, 8572300077865326551637379806655621846104], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [3, 16, 1, 1], [3, 16, 2, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 12, 1, 1], [4, 24, 1, 2], [4, 36, 1, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2], [6, 6, 1, 2], [6, 6, 2, 2], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 1, 1], [6, 16, 2, 1], [8, 36, 2, 2], [12, 2, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [12, 6, 2, 1], [12, 12, 1, 1], [12, 12, 2, 2], [12, 16, 1, 1], [12, 16, 2, 2], [12, 16, 4, 1], [12, 24, 2, 4], [12, 24, 4, 2], [12, 36, 2, 1], [12, 48, 2, 1], [12, 48, 4, 1], [24, 36, 4, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5.D_6^2', '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': '576.8659', 'autcentquo_hash': 8659, 'autcentquo_nilpotent': False, 'autcentquo_order': 576, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^2:D_6^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 1, 2], [3, 2, 3], [3, 8, 3], [3, 16, 3], [4, 2, 1], [4, 6, 1], [4, 12, 1], [4, 24, 2], [4, 36, 1], [6, 1, 2], [6, 2, 3], [6, 3, 4], [6, 6, 6], [6, 8, 3], [6, 16, 3], [8, 36, 4], [12, 2, 2], [12, 4, 3], [12, 6, 2], [12, 12, 5], [12, 16, 9], [12, 24, 16], [12, 36, 2], [12, 48, 6], [24, 36, 8]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '288.857', 'commutator_count': 2, 'commutator_label': '144.155', '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', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 31928, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['576.5071', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [3, 16, 1, 1], [3, 16, 2, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 12, 1, 1], [4, 24, 1, 2], [4, 36, 1, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2], [6, 6, 1, 2], [6, 6, 2, 2], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 1, 1], [6, 16, 2, 1], [8, 36, 2, 2], [12, 2, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [12, 6, 2, 1], [12, 12, 1, 1], [12, 12, 2, 2], [12, 16, 1, 1], [12, 16, 2, 2], [12, 16, 4, 1], [12, 24, 2, 8], [12, 36, 2, 1], [12, 48, 2, 3], [24, 36, 4, 2]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 72, 'exponent': 24, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 2]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '432.745', 'hash': 31928, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 3, 2, 6], 'inner_gens': [[3200423, 29280443, 13489309, 40007587, 890233], [17762051, 43332947, 13489309, 40007587, 894937], [3054011, 43779239, 742666, 39714763, 890233], [3200423, 43332947, 25932628, 40007587, 25487491], [28094033, 18444125, 25639804, 40007587, 25487491]], 'inner_hash': 857, 'inner_nilpotent': False, 'inner_order': 288, 'inner_split': True, 'inner_tex': 'D_6:S_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 12], [2, 33], [3, 12], [4, 18], [6, 21], [12, 3]], 'label': '1728.31928', 'linC_count': 928, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 16, 'linQ_dim': 11, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3*C6.GL(2,Z/4)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 54, 'number_characteristic_subgroups': 50, 'number_conjugacy_classes': 99, 'number_divisions': 57, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 354, 'number_subgroup_classes': 374, 'number_subgroups': 2264, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 7], [3, 80], [4, 104], [6, 128], [8, 144], [12, 976], [24, 288]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [592705, 592705, 592705, 592705], 'outer_gens': [[3200423, 43332947, 13489309, 40007587, 890233], [3200423, 43332947, 742666, 40007587, 25492195], [3200423, 15055523, 742666, 40007587, 25487491], [3200423, 43332947, 13489309, 47120047, 890233]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 9], [3, 4], [4, 16], [6, 7], [8, 6], [12, 8], [16, 1], [24, 2]], 'representations': {'PC': {'code': 264949272135441105768142379764828868667014862473590975450645, 'gens': [1, 2, 5, 6, 8], 'pres': [9, 2, 2, 2, 2, 3, 2, 3, 2, 3, 72, 253, 46, 326, 74, 1444, 733, 3605, 158, 31111, 57040, 1987, 214, 46673]}, 'GLZN': {'d': 2, 'p': 84, 'gens': [3200423, 14817625, 892585, 595057, 25782667, 22080046, 17400613, 637573, 7705165]}, 'Perm': {'d': 26, 'gens': [7605914577989905, 1266748834508038216265880, 53523844179886080000, 21136758688690584, 27924137878061520, 646300418472124416000000, 3, 15537062060069870960640000, 32263316890128450846720000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_6.\\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 144, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 2, 3, 4, 2, 12, 2, 3, 12, 2, 2], 'aut_gens': [[3200423, 43332947, 742666, 40007587, 25487491], [27947621, 18883361, 25932628, 46827223, 890233], [3200423, 15055523, 742666, 40007587, 25492195], [3054011, 43779239, 742666, 39714763, 890233], [1546550, 27799226, 13189429, 22229965, 25492195], [28094033, 18439421, 1042546, 40007587, 25487491], [49517021, 4382129, 38379391, 47120047, 890233], [28094033, 18439421, 25639804, 40007587, 25487491], [3200423, 43337651, 742666, 40007587, 25487491], [26440160, 17125916, 38086567, 15117505, 25487491], [3200423, 43332947, 13489309, 47120047, 890233], [3200423, 43332947, 13489309, 47120047, 894937]], 'aut_group': '4608.lj', 'aut_hash': 3301011005301566777, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4608, 'aut_permdeg': 38, 'aut_perms': [355156898985373875270217154944527887631083916, 9178808122487176061880041086544783666064, 395719148671277575949775647747933550542986122, 507384920426145018326208072519368913835020973, 510013489425694822888677562393233702577716633, 255392521522775702417405120873680163374102578, 510013488364665702150430109762316140687951397, 5688177853474116378478890976342955512344, 467173995632557618329197725699906544355746150, 1587319532413219498617016401107210772, 8572300077865326551637379806655621846104], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [3, 16, 1, 1], [3, 16, 2, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 12, 1, 1], [4, 24, 1, 2], [4, 36, 1, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2], [6, 6, 1, 2], [6, 6, 2, 2], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 1, 1], [6, 16, 2, 1], [8, 36, 2, 2], [12, 2, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [12, 6, 2, 1], [12, 12, 1, 1], [12, 12, 2, 2], [12, 16, 1, 1], [12, 16, 2, 2], [12, 16, 4, 1], [12, 24, 2, 4], [12, 24, 4, 2], [12, 36, 2, 1], [12, 48, 2, 1], [12, 48, 4, 1], [24, 36, 4, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5.D_6^2', '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': '576.8659', 'autcentquo_hash': 8659, 'autcentquo_nilpotent': False, 'autcentquo_order': 576, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^2:D_6^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 1, 2], [3, 2, 3], [3, 8, 3], [3, 16, 3], [4, 2, 1], [4, 6, 1], [4, 12, 1], [4, 24, 2], [4, 36, 1], [6, 1, 2], [6, 2, 3], [6, 3, 4], [6, 6, 6], [6, 8, 3], [6, 16, 3], [8, 36, 4], [12, 2, 2], [12, 4, 3], [12, 6, 2], [12, 12, 5], [12, 16, 9], [12, 24, 16], [12, 36, 2], [12, 48, 6], [24, 36, 8]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '288.857', 'commutator_count': 2, 'commutator_label': '144.155', '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', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 31928, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['576.5071', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [3, 16, 1, 1], [3, 16, 2, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 12, 1, 1], [4, 24, 1, 2], [4, 36, 1, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2], [6, 6, 1, 2], [6, 6, 2, 2], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 1, 1], [6, 16, 2, 1], [8, 36, 2, 2], [12, 2, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [12, 6, 2, 1], [12, 12, 1, 1], [12, 12, 2, 2], [12, 16, 1, 1], [12, 16, 2, 2], [12, 16, 4, 1], [12, 24, 2, 8], [12, 36, 2, 1], [12, 48, 2, 3], [24, 36, 4, 2]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 72, 'exponent': 24, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 2]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '432.745', 'hash': 31928, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 3, 2, 6], 'inner_gens': [[3200423, 29280443, 13489309, 40007587, 890233], [17762051, 43332947, 13489309, 40007587, 894937], [3054011, 43779239, 742666, 39714763, 890233], [3200423, 43332947, 25932628, 40007587, 25487491], [28094033, 18444125, 25639804, 40007587, 25487491]], 'inner_hash': 857, 'inner_nilpotent': False, 'inner_order': 288, 'inner_split': True, 'inner_tex': 'D_6:S_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 12], [2, 33], [3, 12], [4, 18], [6, 21], [12, 3]], 'label': '1728.31928', 'linC_count': 928, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 16, 'linQ_dim': 11, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3*C6.GL(2,Z/4)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 54, 'number_characteristic_subgroups': 50, 'number_conjugacy_classes': 99, 'number_divisions': 57, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 354, 'number_subgroup_classes': 374, 'number_subgroups': 2264, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 7], [3, 80], [4, 104], [6, 128], [8, 144], [12, 976], [24, 288]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [592705, 592705, 592705, 592705], 'outer_gens': [[3200423, 43332947, 13489309, 40007587, 890233], [3200423, 43332947, 742666, 40007587, 25492195], [3200423, 15055523, 742666, 40007587, 25487491], [3200423, 43332947, 13489309, 47120047, 890233]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 9], [3, 4], [4, 16], [6, 7], [8, 6], [12, 8], [16, 1], [24, 2]], 'representations': {'PC': {'code': 264949272135441105768142379764828868667014862473590975450645, 'gens': [1, 2, 5, 6, 8], 'pres': [9, 2, 2, 2, 2, 3, 2, 3, 2, 3, 72, 253, 46, 326, 74, 1444, 733, 3605, 158, 31111, 57040, 1987, 214, 46673]}, 'GLZN': {'d': 2, 'p': 84, 'gens': [3200423, 14817625, 892585, 595057, 25782667, 22080046, 17400613, 637573, 7705165]}, 'Perm': {'d': 26, 'gens': [7605914577989905, 1266748834508038216265880, 53523844179886080000, 21136758688690584, 27924137878061520, 646300418472124416000000, 3, 15537062060069870960640000, 32263316890128450846720000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_6.\\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 144, 'wreath_data': None, 'wreath_product': False}
