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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '192.462', 'ambient_counter': 462, 'ambient_order': 192, 'ambient_tex': 'C_{16}:D_6', 'central': True, 'central_factor': False, 'centralizer_order': 192, 'characteristic': False, 'core_order': 2, 'counter': 86, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '192.462.96.a1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '96.a1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '96.7', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 7, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 96, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{48}:C_2', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '192.462', 'aut_centralizer_order': 1536, 'aut_label': '96.a1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '1.1', 'aut_weyl_index': 3072, 'centralizer': '1.a1.a1', 'complements': ['2.d1.a1', '2.d1.d1', '2.d1.b1', '2.d1.c1'], 'conjugacy_class_count': 1, 'contained_in': ['32.a1.b1', '48.a1.a1', '48.d1.b1'], 'contains': ['192.a1.a1'], 'core': '96.a1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [98], 'label': '192.462.96.a1.b1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '96.a1.b1', 'normal_contained_in': ['32.a1.b1', '48.a1.a1'], 'normal_contains': ['192.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '96.a1.b1', 'projective_image': '96.7', 'quotient_action_image': '1.1', 'quotient_action_kernel': '96.7', 'quotient_action_kernel_order': 96, 'quotient_fusion': None, 'short_label': '96.a1.b1', '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': '2.1', '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': [[1]], '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], [2, 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], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, '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': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 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': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': 24, 'aut_gen_orders': [4, 12, 12, 2, 24, 2, 4, 4], 'aut_gens': [[1, 2, 4], [137, 98, 142], [19, 98, 76], [57, 98, 30], [49, 2, 126], [105, 2, 150], [137, 2, 190], [115, 98, 46], [99, 2, 116]], 'aut_group': '3072.bkm', 'aut_hash': 6561289249819409112, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3072, 'aut_permdeg': 64, 'aut_perms': [2735521609385695706245111698050975908907992100322296818790813590789859376958017246778639, 13769930673492949019515930386785556209044350328454692277364550508830915287441382297625189, 79623909497620370788815559473836795133290573436192279296111433665280614189152574594926639, 21009279348225729917702646063042134144715356584404084526390808409892036698247695128729609, 116027577558272950621900653908075681918587520247693803043330446303829175608417193647278489, 2011865329101406107812880488842148966659048275653122273396028837262141055043436356528167, 55274819908432449408962036223120842196141768410202113392600770130395483727173290340556788, 52608603998704793416046826138093380775116979657655213982375762871625611983530429595533900], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 24, 2, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 24, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [8, 2, 2, 2], [12, 2, 2, 2], [16, 2, 8, 1], [24, 2, 4, 2], [48, 2, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4.D_4^2:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': 96, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 24, 2], [3, 2, 1], [4, 2, 2], [4, 24, 2], [6, 2, 3], [8, 2, 4], [12, 2, 4], [16, 2, 8], [24, 2, 8], [48, 2, 16]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '48.7', 'commutator_count': 1, 'commutator_label': '24.2', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 462, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['96.7', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 24, 1, 2], [3, 2, 1, 1], [4, 2, 1, 2], [4, 24, 1, 2], [6, 2, 1, 3], [8, 2, 2, 2], [12, 2, 2, 2], [16, 2, 4, 2], [24, 2, 4, 2], [48, 2, 8, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 672, 'exponent': 48, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '24.14', 'hash': 462, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 1, 24], 'inner_gens': [[1, 2, 92], [1, 2, 4], [105, 2, 4]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': None, 'inner_tex': 'D_{24}', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 46]], 'label': '192.462', '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': 'C16:D6', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 18, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 54, 'number_divisions': 24, 'number_normal_subgroups': 39, 'number_subgroup_autclasses': 64, 'number_subgroup_classes': 90, 'number_subgroups': 392, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 51], [3, 2], [4, 52], [6, 6], [8, 8], [12, 8], [16, 16], [24, 16], [48, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 2, 4, 2], 'outer_gen_pows': [0, 0, 144, 0, 0], 'outer_gens': [[3, 2, 102], [3, 98, 148], [3, 98, 28], [1, 2, 78], [3, 2, 28]], 'outer_group': '64.263', 'outer_hash': 263, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 32, 'outer_perms': [75478649140557418206773440476467161, 94771829039958036852978353993392420, 111849551449397062840930155182164315, 254906733510959884143548193122684969, 111849549820722256169101240916907234], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4:C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 4], [8, 4], [16, 2]], 'representations': {'PC': {'code': 32061892314217967342341053953, 'gens': [1, 2, 3], 'pres': [7, -2, -2, -2, -2, -2, -2, -3, 1934, 58, 5155, 80, 6164, 102, 6725, 124, 6278]}, 'GLZN': {'d': 2, 'p': 21, 'gens': [9409, 39958, 120406, 164434, 74096, 43861, 38503]}, 'GLZq': {'d': 2, 'q': 64, 'gens': [15010825, 5539357, 8126495, 7370083, 12978193, 1064700, 8650785]}, 'Perm': {'d': 21, 'gens': [534305767728222007, 1, 3068247918569259600, 5366385895345140960, 8061811722467285040, 10637199259039705680, 30]}}, 'schur_multiplier': [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_{16}:D_6', 'transitive_degree': 96, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 24, 'aut_gen_orders': [2, 2, 4, 24], 'aut_gens': [[1, 2], [1, 34], [37, 62], [25, 10], [69, 2]], 'aut_group': '384.12439', 'aut_hash': 12439, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 15, 'aut_perms': [127, 490654080, 7983502, 382414234440], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 24, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 24, 1, 1], [6, 2, 1, 1], [8, 2, 2, 1], [12, 2, 2, 1], [16, 2, 4, 1], [24, 2, 4, 1], [48, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_8:C_4\\times S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '96.209', 'autcentquo_hash': 209, 'autcentquo_nilpotent': False, 'autcentquo_order': 96, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 24, 1], [3, 2, 1], [4, 2, 1], [4, 24, 1], [6, 2, 1], [8, 2, 2], [12, 2, 2], [16, 2, 4], [24, 2, 4], [48, 2, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '48.7', 'commutator_count': 1, 'commutator_label': '24.2', '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': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 24, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 24, 1, 1], [6, 2, 1, 1], [8, 2, 2, 1], [12, 2, 2, 1], [16, 2, 4, 1], [24, 2, 4, 1], [48, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 48, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 8]], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '12.4', 'hash': 7, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 24], 'inner_gens': [[1, 46], [53, 2]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': True, 'inner_tex': 'D_{24}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 23]], 'label': '96.7', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 2, 'linQ_dim': 10, 'linQ_dim_count': 2, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C48:C2', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, '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': 12, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 27, 'number_divisions': 12, 'number_normal_subgroups': 13, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 26, 'number_subgroups': 102, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 25], [3, 2], [4, 26], [6, 2], [8, 4], [12, 4], [16, 8], [24, 8], [48, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 14], [1, 74]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 2], [8, 2], [16, 1]], 'representations': {'PC': {'code': 494470886142670201225089, 'gens': [1, 2], 'pres': [6, -2, -2, -2, -2, -2, -3, 553, 31, 1658, 50, 2115, 69, 2404, 88, 2309]}, 'GLFp': {'d': 2, 'p': 23, 'gens': [267675, 54054]}, 'Perm': {'d': 19, 'gens': [1579934867658601, 8270103144020424, 14273808653392800, 21450439555414824, 28277203544233224, 3]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{48}:C_2', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
