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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '960.2821', 'ambient_counter': 2821, 'ambient_order': 960, 'ambient_tex': '(C_2\\times D_{60}):C_4', 'central': False, 'central_factor': False, 'centralizer_order': 480, 'characteristic': True, 'core_order': 12, 'counter': 233, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '960.2821.80.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '80.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '80.37', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 37, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 80, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times D_{20}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '12.5', 'subgroup_hash': 5, 'subgroup_order': 12, 'subgroup_tex': 'C_2\\times C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.2821', 'aut_centralizer_order': 7680, 'aut_label': '80.a1', 'aut_quo_index': 4, 'aut_stab_index': 1, 'aut_weyl_group': '4.2', 'aut_weyl_index': 7680, 'centralizer': '2.d1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.a1.a1', '40.a1.a1', '40.b1.a1', '40.c1.a1', '40.h1.a1', '40.h1.b1', '40.i1.a1', '40.i1.b1'], 'contains': ['160.a1.a1', '160.b1.a1', '240.a1.a1'], 'core': '80.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6418, 5759, 6890, 5562, 6437, 8278, 6470, 6068], 'generators': [482, 8, 640], 'label': '960.2821.80.a1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '80.a1.a1', 'normal_contained_in': ['16.a1.a1', '40.a1.a1', '40.b1.a1', '40.c1.a1'], 'normal_contains': ['160.a1.a1', '240.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '80.a1.a1', 'projective_image': '480.616', 'quotient_action_image': '2.1', 'quotient_action_kernel': '40.13', 'quotient_action_kernel_order': 40, 'quotient_fusion': None, 'short_label': '80.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [6, 9], [6, 11]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 31], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [6, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.4', 'autcent_hash': 4, 'autcent_nilpotent': False, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_6', '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], [6, 1, 6]], 'center_label': '12.5', 'center_order': 12, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [6, 1, 2, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '12.5', 'hash': 5, '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, 12]], 'label': '12.5', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 6, 'linQ_dim': 3, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C6', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 12, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 9], [6, 11]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4]], 'representations': {'PC': {'code': 273, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16507]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 2064]}, 'Perm': {'d': 7, 'gens': [720, 24, 4]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], '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_6', 'transitive_degree': 12, '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': [2, 2, 6, 20, 10, 12, 12, 4, 12], 'aut_gens': [[1, 4, 16, 32], [641, 4, 280, 928], [651, 12, 280, 936], [327, 4, 150, 616], [643, 4, 278, 360], [173, 4, 950, 360], [643, 12, 272, 232], [655, 4, 88, 232], [809, 12, 62, 744], [329, 12, 638, 424]], 'aut_group': None, 'aut_hash': 8929677708077056321, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 30720, 'aut_permdeg': 128, 'aut_perms': [298894365251342095769653997653982214576907986400006056697923521862924877034003468777936036579438268858528604481969013217100490126208817674943074923493144908519864136230554941743933617660238647928497567164939397713202, 190250057966099295390695040710532282190403255380654899367131729175527847229465408273844291046939204710304035367755914674004499171718215099517339230335554082517280806078700669077113320446963757780232161027303269123395, 24757004298074651930065753255823583350981697581741021505117510235477166987545661508433941844472865844453654180861617612826642468865796632689433487379871970655281235133186348291259484807283974000929857741304442924414, 329803205499072949479688784349570629570493125572315745413147653122545165592693194233117742435913232904535452320731815568501435039265623595561676559775746916025116343931135753488898798230205999690759228692436515223219, 40177485214508191904192712614293043495257867694811784549574490825926518785204400292952631423063991301883882511325939306318894692449785868710427446627756640064893385677392361616308623423601847672394549094454781947166, 329798499276403020346405131940221486199128967240839097038147927243097873050113000399558542473491449739876782984762394392221354990219473644629902012627546801737076266283650961081811834460167347848905899420448351226834, 130508997131030212427821518353249945962850127503315089342965061619197420688963456820025135352598301674741468068962475795046872910356428259574160324320000122283188749987426124585374908004027170475316028411597267391956, 123780343104782326182119335799947596933559775519059947419598071049027952038919263832027058881288497568122869591983638542756486064682206437283181457385625119348949436625283455845520971105745977944142518228611795144355, 383980297354139494867248942532439798953968924047618363807444447104535481145356111255652917124243784290525579938782324177754767167235547095399754627772743955116116291422460447913964596175812302317497401301218359307576], 'aut_phi_ratio': 120.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 60, 2, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 3], [4, 12, 4, 1], [4, 20, 4, 1], [4, 60, 2, 1], [5, 2, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 1, 2], [10, 2, 2, 1], [10, 2, 4, 1], [10, 4, 2, 2], [12, 2, 2, 2], [12, 4, 1, 2], [12, 20, 8, 1], [15, 4, 2, 1], [20, 2, 4, 2], [20, 4, 2, 2], [20, 12, 16, 1], [30, 4, 2, 1], [30, 4, 4, 3], [60, 4, 4, 4]], 'aut_supersolvable': True, 'aut_tex': 'C_{15}:(C_2\\times C_4\\times C_2^2.C_2^6)', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '1920.236168', 'autcentquo_hash': 3337138837734452234, 'autcentquo_nilpotent': False, 'autcentquo_order': 1920, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3\\times D_6\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 60, 2], [3, 2, 1], [4, 1, 2], [4, 2, 3], [4, 12, 4], [4, 20, 4], [4, 60, 2], [5, 2, 2], [6, 2, 3], [6, 4, 2], [10, 2, 6], [10, 4, 4], [12, 2, 4], [12, 4, 2], [12, 20, 8], [15, 4, 2], [20, 2, 8], [20, 4, 4], [20, 12, 16], [30, 4, 14], [60, 4, 16]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '240.28', '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': 2821, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 60, 1, 2], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 3], [4, 12, 2, 2], [4, 20, 2, 2], [4, 60, 1, 2], [5, 2, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 1, 2], [10, 2, 2, 1], [10, 2, 4, 1], [10, 4, 2, 2], [12, 2, 2, 2], [12, 4, 1, 2], [12, 20, 4, 2], [15, 4, 2, 1], [20, 2, 4, 2], [20, 4, 2, 2], [20, 12, 8, 2], [30, 4, 2, 1], [30, 4, 4, 3], [60, 4, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 8064, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 16]], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '120.42', 'hash': 2821, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [4, 1, 2, 30], 'inner_gens': [[1, 4, 496, 360], [1, 4, 16, 32], [481, 4, 16, 928], [649, 4, 80, 32]], 'inner_hash': 28, 'inner_nilpotent': False, 'inner_order': 240, 'inner_split': False, 'inner_tex': 'D_{30}:C_4', 'inner_used': [1, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 64, 'irrQ_dim': 64, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 52], [4, 46]], 'label': '960.2821', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 16, 'linQ_dim': 14, 'linQ_dim_count': 16, 'linR_count': 8, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*D60):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': 42, 'number_characteristic_subgroups': 76, 'number_conjugacy_classes': 114, 'number_divisions': 48, 'number_normal_subgroups': 96, 'number_subgroup_autclasses': 232, 'number_subgroup_classes': 316, 'number_subgroups': 2216, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 127], [3, 2], [4, 256], [5, 4], [6, 14], [10, 28], [12, 176], [15, 8], [20, 224], [30, 56], [60, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 480, 480, 0], 'outer_gens': [[1, 4, 16, 352], [1, 12, 16, 32], [3, 4, 16, 40], [5, 4, 16, 32], [5, 4, 30, 32], [1, 4, 16, 224]], 'outer_group': '128.2319', 'outer_hash': 2319, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 14, 'outer_perms': [127, 6227020800, 39916800, 362880, 5040, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 10], [8, 15], [16, 1], [32, 1], [64, 1]], 'representations': {'PC': {'code': 4505267892367179948927176568563092850364186631, 'gens': [1, 3, 5, 6], 'pres': [8, -2, -2, -2, -2, 2, 2, -3, -5, 16, 66, 19844, 492, 17285, 2821, 141, 39430, 3174, 222, 3111]}, 'Perm': {'d': 24, 'gens': [28260973958679956391144, 28270834328257494624001, 56366109826659886538880, 83483495946041507729280, 6797702856169218470400, 110529224495753440970880, 3, 6504]}}, 'schur_multiplier': [2, 2, 4], '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_2\\times D_{60}):C_4', 'transitive_degree': 240, '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': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 2, 2, 4, 2, 2, 2, 2, 5], 'aut_gens': [[1, 2, 4], [3, 2, 54], [61, 42, 44], [61, 2, 44], [1, 2, 68], [3, 2, 44], [3, 2, 4], [1, 2, 36], [41, 2, 4], [17, 2, 4]], 'aut_group': '1280.1106474', 'aut_hash': 1106474, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1280, 'aut_permdeg': 13, 'aut_perms': [18638650, 766800, 1079215920, 13, 7621200, 1614937680, 23, 486985680, 41], 'aut_phi_ratio': 40.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 10, 4, 1], [4, 2, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1], [10, 2, 4, 1], [20, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2\\times F_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 10, 4], [4, 2, 2], [5, 2, 2], [10, 2, 6], [20, 2, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '20.4', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 37, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['40.6', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 10, 1, 4], [4, 2, 1, 2], [5, 2, 2, 1], [10, 2, 2, 3], [20, 2, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 126, 'exponent': 20, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '40.13', 'hash': 37, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 1, 10], 'inner_gens': [[1, 2, 76], [1, 2, 4], [9, 2, 4]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': False, 'inner_tex': 'D_{10}', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 18]], 'label': '80.37', 'linC_count': 32, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 32, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D20', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, '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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 26, 'number_divisions': 16, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 54, 'number_subgroups': 178, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 43], [4, 4], [5, 4], [10, 12], [20, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 4], 'outer_gen_pows': [0, 20, 0, 0], 'outer_gens': [[1, 42, 44], [21, 42, 4], [21, 42, 6], [1, 2, 68]], 'outer_group': '64.206', 'outer_hash': 206, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 12, 'outer_perms': [207340560, 207340561, 183300497, 262766160], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2:C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 4], [8, 2]], 'representations': {'PC': {'code': 2096911834394627, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -5, 1142, 42, 1443, 58, 1604]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [25038648174890735, 125101729074815868, 91655870133565021]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 14536340629, 58658042680, 68338539124, 113900364736]}, 'Perm': {'d': 11, 'gens': [766087, 720, 4032720, 7983360, 37]}}, '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_{20}', 'transitive_degree': 40, 'wreath_data': None, 'wreath_product': False}
