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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '4608.pc', 'ambient_counter': 393, 'ambient_order': 4608, 'ambient_tex': 'C_2^5.D_6^2', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 16, 'counter': 1364, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4608.pc.72.DQ', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '72.dq1', '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': 72, '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': '64.202', 'subgroup_hash': 202, 'subgroup_order': 64, 'subgroup_tex': 'C_2^3:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4608.pc', 'aut_centralizer_order': None, 'aut_label': '72.DQ', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '576.E', 'complements': None, 'conjugacy_class_count': 3, 'contained_in': ['24.EQ', '24.EV', '36.J', '36.Q', '36.CK'], 'contains': ['144.BX', '144.CE', '144.CX', '144.DB', '144.EF', '144.EK', '144.EM', '144.NS', '144.OO'], 'core': '288.A', 'coset_action_label': None, 'count': 54, 'diagramx': [6356, -1, 3886, -1], 'generators': [127370880, 87181948495, 87091200, 40298900, 5759, 3666193], 'label': '4608.pc.72.DQ', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.I', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.I', 'old_label': '72.dq1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '72.DQ', '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [8, 4], 'aut_gens': [[4001, 643, 2565, 1571, 3993], [517, 2727, 3623, 545, 687], [1957, 1575, 3623, 1571, 1561]], 'aut_group': '12288.dp', 'aut_hash': 1630168457023225010, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 24, 'aut_perms': [187522706853152502407387, 128367454260810836992011], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 1, 4, 1], [2, 2, 12, 1], [2, 4, 2, 1], [4, 4, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^9.S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 8092891664598236742, 'autcent_nilpotent': True, 'autcent_order': 2048, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8.C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 12], [2, 4, 2], [4, 4, 6]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 202, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['32.27', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 12], [2, 4, 1, 2], [4, 4, 1, 6]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 420, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '16.14', 'hash': 202, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 1, 2], 'inner_gens': [[4001, 643, 2565, 1571, 2731], [4001, 643, 2565, 1571, 2699], [4001, 643, 2565, 1571, 3993], [4001, 643, 2565, 1571, 3993], [2691, 1921, 2565, 1571, 3993]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12]], 'label': '64.202', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, '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:D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 28, 'number_divisions': 28, 'number_normal_subgroups': 105, 'number_subgroup_autclasses': 38, 'number_subgroup_classes': 331, 'number_subgroups': 569, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 39], [4, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [513, 513, 513, 513, 513, 513, 513, 513, 513, 513], 'outer_gens': [[1575, 643, 2565, 545, 3993], [2695, 3619, 2565, 1539, 1593], [2723, 675, 2565, 1571, 1981], [4001, 643, 3623, 1571, 3993], [2723, 1921, 3591, 1571, 3993], [1957, 2695, 2565, 1571, 1981], [1957, 643, 2565, 1571, 3993], [4001, 1921, 2565, 1571, 3993], [4001, 1953, 2565, 1571, 3993], [2723, 643, 2565, 1571, 3993]], 'outer_group': '1536.408632814', 'outer_hash': 2972356385187470528, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 14, 'outer_perms': [1683987865, 7878390024, 21317548327, 14390022966, 21797289414, 243538566, 85686, 2438997120, 21316418640, 2440489806], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12]], 'representations': {'PC': {'code': 4564455696, 'gens': [1, 2, 3, 4, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 724, 730, 88]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [705686951426, 141602720900, 706461793378, 706204576418]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [11040833, 31029334, 19896385, 10743875, 6210625, 19208257]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [643, 2565, 523, 545, 1539, 687]}, 'Perm': {'d': 10, 'gens': [367925, 766087, 85800, 806520, 806416, 806400]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3:D_4', 'transitive_degree': 16, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [12, 4, 12, 12, 6, 4, 6], 'aut_gens': [[87261786394, 87185911680, 93529087126, 47537280, 6234660622, 40279680, 127370880, 174404133219, 174356582400, 7286095, 180663824861], [87302033280, 87181952794, 174364221970, 7257600, 180587264672, 40279680, 87091200, 93412585436, 93405312000, 79846675, 6270930960], [87185944474, 87261753600, 174396504556, 43545600, 180667071359, 127370880, 40279680, 93448872697, 93405312000, 123406015, 6314489373], [87178686874, 87218208000, 174440063768, 79833600, 180623525490, 127370880, 87091200, 93485172537, 93405312000, 120127773, 6270956095], [6350795674, 6230649600, 174364203283, 7257600, 87181925759, 40279680, 87091200, 93412584697, 93405312000, 79835165, 180627540175], [87218208000, 87178686874, 93412951448, 3991680, 6227416352, 127370880, 87091200, 174360589654, 174356582400, 83848896, 180703384714], [87261786394, 87185911680, 174360244124, 3991680, 180591256352, 40279680, 127370880, 93409319516, 93405312000, 120150179, 6274196765], [6227383680, 6266970394, 93488807446, 79833600, 180623520269, 127370880, 87091200, 174436425143, 174356582400, 120138461, 87222201360]], 'aut_group': None, 'aut_hash': 8782537470167940265, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 110592, 'aut_permdeg': 72, 'aut_perms': [55442194820628974489710485789331226063559940866922159161797124289659178991707607642323533473321129081600, 24649458073410983690315190435222841709803748267182830170169875923957334579002908279246613643982181032664, 24345876740302966953039993830668495149145833540181408456093033677282049660621943631863185313333999162712, 4858025463776953705319022891145100837734371054404410108134182391837931127029246117338219519613750022840, 52830082884689239068610026803800479526645879850415292056023259095652026482850527604600334223099659690389, 36253351102741682019876125493998165049443796914385932570506428250837881849050560968964244820321005328305, 40988556977494597706684805809382239100084580111179624238932715608105628822919822698401165001681230460508], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [2, 3, 1, 2], [2, 4, 3, 1], [2, 6, 3, 1], [2, 12, 2, 1], [2, 12, 3, 1], [2, 18, 2, 1], [2, 24, 2, 1], [2, 36, 2, 1], [2, 36, 3, 1], [2, 72, 3, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 4, 3, 1], [4, 12, 3, 1], [4, 18, 2, 1], [4, 24, 2, 1], [4, 24, 3, 1], [4, 36, 3, 1], [4, 48, 3, 2], [4, 72, 3, 4], [6, 2, 1, 1], [6, 4, 3, 1], [6, 6, 1, 2], [6, 8, 1, 1], [6, 8, 3, 1], [6, 12, 3, 1], [6, 16, 1, 1], [6, 16, 3, 1], [6, 24, 3, 1], [6, 32, 3, 2], [6, 32, 6, 1], [6, 48, 2, 1], [6, 96, 2, 1], [12, 8, 3, 1], [12, 24, 3, 1], [12, 32, 3, 1], [12, 32, 6, 1], [12, 48, 2, 1], [12, 96, 3, 2], [12, 192, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6.C_3^3.C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '6912.go', 'autcentquo_hash': 530348522389483268, 'autcentquo_nilpotent': False, 'autcentquo_order': 6912, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_6\\times S_4^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 3, 2], [2, 4, 3], [2, 6, 3], [2, 12, 5], [2, 18, 2], [2, 24, 2], [2, 36, 5], [2, 72, 3], [3, 2, 1], [3, 8, 1], [3, 16, 1], [4, 4, 3], [4, 12, 3], [4, 18, 2], [4, 24, 5], [4, 36, 3], [4, 48, 6], [4, 72, 12], [6, 2, 1], [6, 4, 3], [6, 6, 2], [6, 8, 4], [6, 12, 3], [6, 16, 4], [6, 24, 3], [6, 32, 12], [6, 48, 2], [6, 96, 2], [12, 8, 3], [12, 24, 3], [12, 32, 9], [12, 48, 2], [12, 96, 6], [12, 192, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '2304.sm', 'commutator_count': 1, 'commutator_label': '288.1041', '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', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 393, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 3, 1, 2], [2, 4, 1, 3], [2, 6, 1, 3], [2, 12, 1, 5], [2, 18, 1, 2], [2, 24, 1, 2], [2, 36, 1, 5], [2, 72, 1, 3], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 4, 1, 3], [4, 12, 1, 3], [4, 18, 1, 2], [4, 24, 1, 5], [4, 36, 1, 3], [4, 48, 1, 6], [4, 72, 1, 12], [6, 2, 1, 1], [6, 4, 1, 3], [6, 6, 1, 2], [6, 8, 1, 4], [6, 12, 1, 3], [6, 16, 1, 4], [6, 24, 1, 3], [6, 32, 1, 12], [6, 48, 1, 2], [6, 96, 1, 2], [12, 8, 1, 3], [12, 24, 1, 3], [12, 32, 1, 9], [12, 48, 1, 2], [12, 96, 1, 6], [12, 192, 1, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 12, 'exponents_of_order': [9, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 2]], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '576.8659', 'hash': 187372022923601090, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 6, 3, 2, 2, 2, 6, 3, 6, 4], 'inner_gens': [[87261786394, 87185911680, 174356978326, 7257600, 180591243022, 127370880, 40279680, 93412583139, 93405312000, 47565775, 6354053981], [87261786394, 87185911680, 174396532246, 120113280, 180591243022, 127370880, 40279680, 93525438819, 93405312000, 4020175, 6314500061], [6227416474, 6266937600, 93529087126, 120113280, 180623539342, 87091200, 127370880, 174476722617, 174356582400, 79848093, 87225504179], [87181952794, 87302033280, 93445261846, 47537280, 6350782222, 87091200, 40279680, 174404133219, 174356582400, 83853775, 180710636381], [180667098394, 180591223680, 174396532246, 120113280, 6234660622, 127370880, 40279680, 93525438819, 93405312000, 4028579, 87265770461], [87185944474, 87261753600, 93408973846, 79833600, 6310502542, 40279680, 127370880, 174436429539, 174356582400, 83853775, 180630802781], [87185944474, 87261753600, 93529087126, 123379200, 6310502542, 40279680, 127370880, 174479975139, 174356582400, 43574095, 180663824861], [180587264794, 180707345280, 93445247890, 47537280, 87302052622, 87091200, 40279680, 174404133219, 174356582400, 83853775, 6354041875], [180667098394, 180591223680, 93529087126, 47537280, 87185931022, 40279680, 127370880, 174404133219, 174356582400, 7286095, 6307242461], [87218240794, 87178654080, 93488779756, 123379200, 6227389170, 127370880, 87091200, 174479975139, 174356582400, 7286095, 180627524755], [6227416474, 6266937600, 174360230290, 43545600, 87218227342, 87091200, 127370880, 93448882170, 93405312000, 47574179, 180663824861]], 'inner_hash': 4594365871385040477, 'inner_nilpotent': False, 'inner_order': 2304, 'inner_split': True, 'inner_tex': 'C_2^6:S_3^2', 'inner_used': [1, 2, 3, 5, 8, 10], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 16], [2, 28], [3, 16], [4, 32], [6, 20], [8, 4], [12, 10], [16, 1], [24, 2]], 'label': '4608.pc', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'C2^5.D6^2', 'ngens': 11, '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 52, 'number_characteristic_subgroups': 39, 'number_conjugacy_classes': 129, 'number_divisions': 129, 'number_normal_subgroups': 221, 'number_subgroup_autclasses': 3646, 'number_subgroup_classes': 14206, 'number_subgroups': 242370, 'old_label': None, 'order': 4608, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 583], [3, 26], [4, 1464], [6, 902], [12, 1632]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 3], 'outer_gen_pows': [0, 0, 0, 0, 7621119], 'outer_gens': [[87261753600, 87185944474, 174396504320, 120113280, 180591229170, 127370880, 40279680, 93525449850, 93405312000, 3993245, 6314513459], [87261786394, 87185911680, 174480324883, 47537280, 180591243022, 40279680, 127370880, 93452862819, 93405312000, 7286095, 6307242461], [87261753600, 87185944474, 174480324883, 47537280, 180591229170, 40279680, 127370880, 93452862819, 93405312000, 7286095, 6307230355], [87261753600, 87185944474, 174480357526, 47537280, 180591229170, 40279680, 127370880, 93452862819, 93405312000, 7286095, 6307230355], [87261786394, 87185911680, 93445234156, 120113280, 6234647039, 127370880, 40279680, 174476710777, 174356582400, 3993245, 180671094179]], 'outer_group': '48.51', 'outer_hash': 51, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 9, 'outer_perms': [41041, 90720, 24, 131784, 3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_6', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 28], [3, 16], [4, 32], [6, 20], [8, 4], [12, 10], [16, 1], [24, 2]], 'representations': {'PC': {'code': '2538998962112367303064933665773262669362447286981084102787138792064654941120403225799850352387093119316417646849', 'gens': [1, 2, 4, 5, 6, 8, 9], 'pres': [11, -2, -2, -3, -2, 2, 2, -2, -2, 2, -2, -3, 34013, 56, 266, 1587, 542, 421, 1005, 191, 106133, 5956, 1215, 192, 4638, 190088, 104563, 9171, 2846, 294, 422409, 211220, 8864, 328, 371722, 185877, 7809]}, 'Perm': {'d': 15, 'gens': [87261786394, 87185911680, 93529087126, 47537280, 6234660622, 40279680, 127370880, 174404133219, 174356582400, 7286095, 180663824861]}}, 'schur_multiplier': None, 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^5.D_6^2', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': None}
