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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1944.2428', 'ambient_counter': 2428, 'ambient_order': 1944, 'ambient_tex': '(A_4\\times \\He_3).S_3', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 12, 'counter': 77, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1944.2428.27.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '27.f1.a1', 'outer_equivalence': False, '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': 27, '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': '72.8', 'subgroup_hash': 8, 'subgroup_order': 72, 'subgroup_tex': 'C_9:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1944.2428', 'aut_centralizer_order': 12, 'aut_label': '27.f1', 'aut_quo_index': None, 'aut_stab_index': 27, 'aut_weyl_group': '216.111', 'aut_weyl_index': 324, 'centralizer': '972.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['9.e1.a1', '9.g1.a1', '9.h1.a1', '9.h1.b1'], 'contains': ['54.f1.a1', '54.i1.a1', '54.n1.a1', '81.b1.a1'], 'core': '162.a1.a1', 'coset_action_label': None, 'count': 27, 'diagramx': [3107, -1, 9047, -1, 6319, -1, 5162, -1], 'generators': [3, 648, 972, 54, 864], 'label': '1944.2428.27.f1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.f1.a1', 'old_label': '27.f1.a1', 'projective_image': '1944.2428', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '27.f1.a1', 'subgroup_fusion': None, 'weyl_group': '36.4'}
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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': 18, 'aut_gen_orders': [2, 6, 18], 'aut_gens': [[380, 123463, 68592], [6840, 123463, 68592], [380, 6877, 96041], [3572, 6877, 68592]], 'aut_group': '216.111', 'aut_hash': 111, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 216, 'aut_permdeg': 13, 'aut_perms': [7, 136211776, 609650063], 'aut_phi_ratio': 9.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 18, 1, 1], [3, 2, 1, 1], [4, 18, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [9, 2, 3, 1], [18, 2, 3, 1], [18, 2, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{18}:C_6', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '54.6', 'autcentquo_hash': 6, 'autcentquo_nilpotent': False, 'autcentquo_order': 54, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_9:C_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 18, 1], [3, 2, 1], [4, 18, 1], [6, 2, 3], [9, 2, 3], [18, 2, 9]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '36.4', 'commutator_count': 1, 'commutator_label': '18.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 8, '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, 1], [2, 18, 1, 1], [3, 2, 1, 1], [4, 18, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [9, 2, 3, 1], [18, 2, 3, 1], [18, 2, 6, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 6, 'exponent': 36, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 6]], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '12.4', 'hash': 8, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 2, 9], 'inner_gens': [[380, 6877, 13728], [6840, 123463, 68592], [1539, 123463, 68592]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'D_{18}', 'inner_used': [1, 2, 3], 'irrC_degree': 2, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 17]], 'label': '72.8', 'linC_count': 6, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 9, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C9:D4', 'ngens': 3, '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': 12, 'number_conjugacy_classes': 21, 'number_divisions': 11, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 24, 'number_subgroups': 80, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 21], [3, 2], [4, 18], [6, 6], [9, 6], [18, 18]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [6860], 'outer_gens': [[380, 123463, 96041]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 1], [6, 2], [12, 1]], 'representations': {'PC': {'code': 15138792414595, 'gens': [1, 2, 3], 'pres': [5, -2, -2, 2, -3, -3, 1052, 42, 1283, 78, 1204]}, 'GLFp': {'d': 2, 'p': 19, 'gens': [380, 123463, 68592]}, 'Perm': {'d': 13, 'gens': [94434607, 5, 16, 609725520, 1133733000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9:D_4', 'transitive_degree': 36, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [12, 6, 18, 6], 'aut_gens': [[1, 6, 18, 108], [1171, 1362, 1314, 1494], [1, 330, 738, 1260], [1811, 384, 360, 126], [1727, 1626, 1710, 1818]], 'aut_group': None, 'aut_hash': 5921079441208705547, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 69984, 'aut_permdeg': 180, 'aut_perms': [130938854382021350345592683510403740656137208080239170536763135516138589589350544344496869963762000590828401981370447592498171874974498351569453803959775122583893893729706996406547985698813680458581736456112931081445015420121817511491507235379841905028978140684755380881255281700513333733773043559496058859194502625215966196186904, 923361083351691072442003640636776543990372730740611020094671791217657035674333805822235512850826052880527084790759217748782765041863377094184201621002909030486412059387012620403993763886257055021452872304391167851707188692119558792942773681593203470981319597943012054270303329267322768101357088935231325637379132373886088219755, 173046783057504712710068895306029993567865861108773173573286841254361740248849426414957390496409981978427464862832099963332256254660118823725582724325683261403914181092306744395467212498320974917738225653344944438901619407733106470728780618118161635548656605345319491426070867046139177533440374464783492936846820622036802021565857, 104769671857019815282154245609745209336055521473787166916488457756653940955273163933428364436630922246705380364832194686370298932693002680472789977647925781045822654762006289526137593758467785531701310928530235006848151927402622779731307752153729690816766547870031725266579988915381043393028672162507516055707945398225950018311481], 'aut_phi_ratio': 108.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 162, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 8, 3, 1], [3, 9, 2, 1], [3, 18, 2, 1], [3, 24, 2, 1], [3, 72, 2, 1], [3, 72, 4, 1], [4, 162, 1, 1], [6, 6, 1, 1], [6, 18, 1, 1], [6, 27, 2, 1], [6, 54, 2, 1], [6, 162, 2, 1], [9, 6, 3, 1], [9, 24, 6, 1], [12, 162, 2, 1], [18, 18, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.C_3^5.C_2^3', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 5921079441208705547, 'autcentquo_nilpotent': False, 'autcentquo_order': 69984, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.C_3^5.C_2^3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 162, 1], [3, 2, 1], [3, 6, 1], [3, 8, 3], [3, 9, 2], [3, 18, 2], [3, 24, 2], [3, 72, 6], [4, 162, 1], [6, 6, 1], [6, 18, 1], [6, 27, 2], [6, 54, 2], [6, 162, 2], [9, 6, 3], [9, 24, 6], [12, 162, 2], [18, 18, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1944.2428', 'commutator_count': 1, 'commutator_label': '324.126', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2428, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 162, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 8, 1, 3], [3, 9, 2, 1], [3, 18, 2, 1], [3, 24, 1, 2], [3, 72, 2, 3], [4, 162, 1, 1], [6, 6, 1, 1], [6, 18, 1, 1], [6, 27, 2, 1], [6, 54, 2, 1], [6, 162, 2, 1], [9, 6, 3, 1], [9, 24, 3, 2], [12, 162, 2, 1], [18, 18, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 49140, 'exponent': 36, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[18, 1, 3]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '216.164', 'hash': 2428, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 3, 6, 18], 'inner_gens': [[1, 12, 1386, 630], [13, 6, 990, 1134], [685, 978, 18, 108], [1531, 1032, 18, 108]], 'inner_hash': 2428, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': True, 'inner_tex': '(A_4\\times \\He_3).S_3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 18, 'irrQ_degree': 54, 'irrQ_dim': 54, 'irrR_degree': 18, 'irrep_stats': [[1, 6], [2, 12], [3, 6], [6, 15], [18, 4]], 'label': '1944.2428', 'linC_count': 54, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 21, 'linQ_degree_count': 6, 'linQ_dim': 21, 'linQ_dim_count': 6, 'linR_count': 18, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': '(A4*He3).S3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 21, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 43, 'number_divisions': 26, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 171, 'number_subgroup_classes': 237, 'number_subgroups': 5980, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 165], [3, 566], [4, 162], [6, 510], [9, 162], [12, 324], [18, 54]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 6, 90, 540], [5, 1302, 90, 108]], 'outer_group': '36.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 36, 'outer_permdeg': 8, 'outer_perms': [5761, 28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times S_3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 31, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [3, 2], [4, 4], [6, 6], [12, 1], [18, 4], [54, 1]], 'representations': {'PC': {'code': 16002464198789120165941048043950149912227333364102397896865835371, 'gens': [1, 3, 4, 6], 'pres': [8, 2, 3, 3, 2, 3, 2, 3, 3, 16, 290, 44355, 21035, 5299, 91, 28804, 13692, 30245, 34573, 9093, 141, 64518, 26222, 222, 82951]}, 'Perm': {'d': 31, 'gens': [1612521832071790617710790758525, 293651207864754572426801797215360, 3, 568086624299121505568637479885640, 44528286387336810840699950204640, 842497740714314660519445106072320, 7, 16]}}, 'schur_multiplier': [3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(A_4\\times \\He_3).S_3', 'transitive_degree': 108, 'wreath_data': None, 'wreath_product': False}
