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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '384.4615', 'ambient_counter': 4615, 'ambient_order': 384, 'ambient_tex': '(C_6\\times D_8):C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 48, 'counter': 14, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '384.4615.4.g1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '4.g1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 4, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '96.159', 'subgroup_hash': 159, 'subgroup_order': 96, 'subgroup_tex': 'C_2^3.D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.4615', 'aut_centralizer_order': None, 'aut_label': '4.g1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '48.a1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['2.a1'], 'contains': ['8.b1', '8.h1', '8.i1', '8.j1', '12.d1'], 'core': '8.b1', 'coset_action_label': None, 'count': 8, 'diagramx': None, 'generators': [1, 128, 8, 4, 192, 2], 'label': '384.4615.4.g1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1', 'old_label': '4.g1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.g1', 'subgroup_fusion': None, 'weyl_group': None}
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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': 3, 'aut_exponent': 24, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2], 'aut_gens': [[1, 4, 8, 16], [7, 4, 8, 88], [9, 4, 8, 90], [3, 4, 8, 84], [3, 4, 8, 88], [9, 4, 8, 88], [3, 4, 8, 80], [61, 4, 8, 94], [33, 4, 8, 16], [11, 12, 8, 88], [11, 4, 8, 88], [1, 6, 8, 80]], 'aut_group': '3072.bkk', 'aut_hash': 4717764161008590725, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3072, 'aut_permdeg': 19, 'aut_perms': [13875745173264796, 6445526960240209, 6449462916387436, 362880, 6446835113609784, 362935, 20717759480522255, 3991680, 6446835113609521, 368209, 6405076233118134], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 1, 4, 1], [2, 2, 4, 1], [3, 2, 1, 1], [4, 6, 8, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [6, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^5.(D_4\\times D_6)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.8935', 'autcent_hash': 8935, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6:C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 4], [3, 2, 1], [4, 6, 8], [6, 2, 15]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', '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': 159, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['48.19', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 4], [3, 2, 1, 1], [4, 6, 2, 4], [6, 2, 1, 7], [6, 2, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 84, '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': '24.14', 'hash': 159, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 1, 1, 6], 'inner_gens': [[1, 4, 8, 88], [1, 4, 8, 16], [1, 4, 8, 16], [41, 4, 8, 16]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': False, 'inner_tex': 'D_6', 'inner_used': [1, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 20]], 'label': '96.159', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'C2^3.D6', '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': 11, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 36, 'number_divisions': 28, 'number_normal_subgroups': 65, 'number_subgroup_autclasses': 44, 'number_subgroup_classes': 132, 'number_subgroups': 242, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 15], [3, 2], [4, 48], [6, 30]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 6, 8, 16], [53, 4, 8, 16], [5, 4, 8, 20], [3, 12, 8, 16], [3, 4, 8, 18], [5, 4, 8, 16], [1, 4, 8, 24], [3, 4, 8, 24]], 'outer_group': '256.3324', 'outer_hash': 3324, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 16, 'outer_perms': [2878327872055, 381378, 4296764104709, 998288209, 82, 12933406080, 1321086787996, 12316], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5:D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 4]], 'representations': {'PC': {'code': 64329483338817, 'gens': [1, 3, 4, 5], 'pres': [6, -2, -2, -2, -2, 2, -3, 12, 2644, 88, 2309]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58301502099033346, 125101750005088212, 25038659797524321]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [1745, 8645, 19019, 1777, 1801, 9587]}, 'Perm': {'d': 13, 'gens': [479042049, 40336, 1037877136, 16, 482630400, 840]}}, 'schur_multiplier': [2, 2, 2, 2], '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_2^3.D_6', 'transitive_degree': 48, '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': 24, 'aut_gen_orders': [6, 4, 4, 2, 4, 4, 12, 2, 4], 'aut_gens': [[1, 4, 16], [321, 132, 306], [321, 36, 274], [97, 108, 16], [195, 14, 122], [291, 198, 280], [241, 292, 178], [323, 230, 218], [9, 206, 88], [139, 238, 274]], 'aut_group': None, 'aut_hash': 8847658690579714531, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 49152, 'aut_permdeg': 80, 'aut_perms': [7044755196328006741748210288697634903617018690374912616528820620770353160866080065577889482800865278059285834405572846, 27228857824190867150550673168788041561369516570462081911766154479102063371917394304150393766091398754544073716486500894, 48566402293704327756798496043325374688258425885581919976609808020283990547018159483450781874775035919813133960733332197, 22503869944651001909295135284610468110077325848559835386216900681561829169691662157017180204522497300941079327797722212, 67627681186176661569015574506208836059654587287815577882676604617792939775767243544240684778922481368619759889836435165, 41147214036722680971302495202666683196104508703000805054398219442680042714044076864462504456478702506319085559552043952, 38984078421588144336038205324700334983414407286187326649979999343138124194191289665208068499587023087339887633665190566, 14270659196106630786677598470414079534449414420082601449532942173702170306285604037945105183733465498497158342199950434, 56766360428947338613597906550929777483559889312691227451477745608562890606205427040329406619073167599180283264125028577], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 3], [2, 8, 4, 1], [3, 2, 1, 1], [4, 2, 1, 4], [4, 12, 4, 1], [4, 24, 4, 1], [6, 2, 1, 1], [6, 2, 2, 3], [6, 8, 8, 1], [8, 4, 4, 1], [8, 12, 4, 1], [12, 4, 1, 4], [24, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:((C_2^6\\times C_4).C_2^6)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '512.10494213', 'autcent_hash': 1718285292446712970, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^9', '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, 7], [2, 8, 4], [3, 2, 1], [4, 2, 4], [4, 12, 4], [4, 24, 4], [6, 2, 7], [6, 8, 8], [8, 4, 4], [8, 12, 4], [12, 4, 4], [24, 4, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '48.38', 'commutator_count': 1, 'commutator_label': '24.9', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4615, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 8, 1, 4], [3, 2, 1, 1], [4, 2, 1, 4], [4, 12, 2, 2], [4, 24, 2, 2], [6, 2, 1, 7], [6, 8, 2, 4], [8, 4, 1, 4], [8, 12, 2, 2], [12, 4, 1, 4], [24, 4, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 336, 'exponent': 24, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.10', 'frattini_quotient': '24.14', 'hash': 4615, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 12], 'inner_gens': [[1, 14, 176], [11, 4, 80], [225, 324, 16]], 'inner_hash': 38, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': False, 'inner_tex': 'S_3\\times D_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 28], [4, 16]], 'label': '384.4615', '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': '(C6*D8):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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 25, 'number_characteristic_subgroups': 43, 'number_conjugacy_classes': 60, 'number_divisions': 46, 'number_normal_subgroups': 113, 'number_subgroup_autclasses': 166, 'number_subgroup_classes': 376, 'number_subgroups': 1278, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 39], [3, 2], [4, 152], [6, 78], [8, 64], [12, 16], [24, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 4, 2, 4, 4, 4, 4], 'outer_gen_pows': [96, 96, 0, 0, 0, 0, 96], 'outer_gens': [[345, 100, 24], [49, 198, 18], [3, 4, 304], [1, 294, 274], [1, 300, 24], [3, 100, 122], [49, 4, 208]], 'outer_group': None, 'outer_hash': 5611842860441815772, 'outer_nilpotent': True, 'outer_order': 1024, 'outer_permdeg': 20, 'outer_perms': [891063873256850056, 121688253750083766, 262581193718288040, 891753017521766526, 391992397841721742, 769418778948382102, 262915958403313921], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6.C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 14], [8, 4]], 'representations': {'PC': {'code': 34902357572210916977014266903813962564915524865, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, 2, -2, -2, -2, -2, -3, 16, 338, 66, 7044, 820, 116, 16901, 1941, 141, 17926, 4502, 166, 16391, 4119]}, 'Perm': {'d': 19, 'gens': [6403855776958825, 6404995361688497, 4103734091507, 127388114, 23895, 4103606707200, 30235, 6758061133824000]}}, 'schur_multiplier': [2, 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_6\\times D_8):C_4', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
