direct product, metabelian, supersoluble, monomial
Aliases: C3×C32⋊11SD16, C33⋊26SD16, C12.34(S3×C6), (Q8×C33)⋊2C2, C12⋊S3.5C6, C32⋊4C8⋊12C6, (C3×C12).129D6, (Q8×C32)⋊12S3, (Q8×C32)⋊13C6, (C32×C6).75D4, C32⋊15(C3×SD16), C6.37(C32⋊7D4), C32⋊14(Q8⋊2S3), (C32×C12).29C22, C4.3(C6×C3⋊S3), Q8⋊3(C3×C3⋊S3), (C3×Q8)⋊5(C3×S3), C12.54(C2×C3⋊S3), (C3×Q8)⋊6(C3⋊S3), C3⋊3(C3×Q8⋊2S3), (C3×C6).71(C3×D4), C6.40(C3×C3⋊D4), (C3×C12).48(C2×C6), (C3×C12⋊S3).5C2, C2.6(C3×C32⋊7D4), (C3×C32⋊4C8)⋊17C2, (C3×C6).110(C3⋊D4), SmallGroup(432,493)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C32⋊11SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d3 >
Subgroups: 564 in 168 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, D4, Q8, C32, C32, C32, C12, C12, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×Q8, C3×Q8, C3×Q8, C33, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, Q8⋊2S3, C3×SD16, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C32⋊4C8, C3×D12, C12⋊S3, Q8×C32, Q8×C32, Q8×C32, C32×C12, C32×C12, C6×C3⋊S3, C3×Q8⋊2S3, C32⋊11SD16, C3×C32⋊4C8, C3×C12⋊S3, Q8×C33, C3×C32⋊11SD16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3⋊D4, C3×D4, S3×C6, C2×C3⋊S3, Q8⋊2S3, C3×SD16, C3×C3⋊S3, C3×C3⋊D4, C32⋊7D4, C6×C3⋊S3, C3×Q8⋊2S3, C32⋊11SD16, C3×C32⋊7D4, C3×C32⋊11SD16
►On 144 points
Generators in S144
(1 47 135)(2 48 136)(3 41 129)(4 42 130)(5 43 131)(6 44 132)(7 45 133)(8 46 134)(9 33 54)(10 34 55)(11 35 56)(12 36 49)(13 37 50)(14 38 51)(15 39 52)(16 40 53)(17 121 120)(18 122 113)(19 123 114)(20 124 115)(21 125 116)(22 126 117)(23 127 118)(24 128 119)(25 59 89)(26 60 90)(27 61 91)(28 62 92)(29 63 93)(30 64 94)(31 57 95)(32 58 96)(65 80 97)(66 73 98)(67 74 99)(68 75 100)(69 76 101)(70 77 102)(71 78 103)(72 79 104)(81 105 138)(82 106 139)(83 107 140)(84 108 141)(85 109 142)(86 110 143)(87 111 144)(88 112 137)
(1 91 124)(2 125 92)(3 93 126)(4 127 94)(5 95 128)(6 121 96)(7 89 122)(8 123 90)(9 88 100)(10 101 81)(11 82 102)(12 103 83)(13 84 104)(14 97 85)(15 86 98)(16 99 87)(17 58 132)(18 133 59)(19 60 134)(20 135 61)(21 62 136)(22 129 63)(23 64 130)(24 131 57)(25 113 45)(26 46 114)(27 115 47)(28 48 116)(29 117 41)(30 42 118)(31 119 43)(32 44 120)(33 112 68)(34 69 105)(35 106 70)(36 71 107)(37 108 72)(38 65 109)(39 110 66)(40 67 111)(49 78 140)(50 141 79)(51 80 142)(52 143 73)(53 74 144)(54 137 75)(55 76 138)(56 139 77)
(1 27 20)(2 21 28)(3 29 22)(4 23 30)(5 31 24)(6 17 32)(7 25 18)(8 19 26)(9 112 75)(10 76 105)(11 106 77)(12 78 107)(13 108 79)(14 80 109)(15 110 73)(16 74 111)(33 137 100)(34 101 138)(35 139 102)(36 103 140)(37 141 104)(38 97 142)(39 143 98)(40 99 144)(41 63 126)(42 127 64)(43 57 128)(44 121 58)(45 59 122)(46 123 60)(47 61 124)(48 125 62)(49 71 83)(50 84 72)(51 65 85)(52 86 66)(53 67 87)(54 88 68)(55 69 81)(56 82 70)(89 113 133)(90 134 114)(91 115 135)(92 136 116)(93 117 129)(94 130 118)(95 119 131)(96 132 120)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)
(1 49)(2 52)(3 55)(4 50)(5 53)(6 56)(7 51)(8 54)(9 46)(10 41)(11 44)(12 47)(13 42)(14 45)(15 48)(16 43)(17 70)(18 65)(19 68)(20 71)(21 66)(22 69)(23 72)(24 67)(25 85)(26 88)(27 83)(28 86)(29 81)(30 84)(31 87)(32 82)(33 134)(34 129)(35 132)(36 135)(37 130)(38 133)(39 136)(40 131)(57 111)(58 106)(59 109)(60 112)(61 107)(62 110)(63 105)(64 108)(73 125)(74 128)(75 123)(76 126)(77 121)(78 124)(79 127)(80 122)(89 142)(90 137)(91 140)(92 143)(93 138)(94 141)(95 144)(96 139)(97 113)(98 116)(99 119)(100 114)(101 117)(102 120)(103 115)(104 118)
G:=sub<Sym(144)| (1,47,135)(2,48,136)(3,41,129)(4,42,130)(5,43,131)(6,44,132)(7,45,133)(8,46,134)(9,33,54)(10,34,55)(11,35,56)(12,36,49)(13,37,50)(14,38,51)(15,39,52)(16,40,53)(17,121,120)(18,122,113)(19,123,114)(20,124,115)(21,125,116)(22,126,117)(23,127,118)(24,128,119)(25,59,89)(26,60,90)(27,61,91)(28,62,92)(29,63,93)(30,64,94)(31,57,95)(32,58,96)(65,80,97)(66,73,98)(67,74,99)(68,75,100)(69,76,101)(70,77,102)(71,78,103)(72,79,104)(81,105,138)(82,106,139)(83,107,140)(84,108,141)(85,109,142)(86,110,143)(87,111,144)(88,112,137), (1,91,124)(2,125,92)(3,93,126)(4,127,94)(5,95,128)(6,121,96)(7,89,122)(8,123,90)(9,88,100)(10,101,81)(11,82,102)(12,103,83)(13,84,104)(14,97,85)(15,86,98)(16,99,87)(17,58,132)(18,133,59)(19,60,134)(20,135,61)(21,62,136)(22,129,63)(23,64,130)(24,131,57)(25,113,45)(26,46,114)(27,115,47)(28,48,116)(29,117,41)(30,42,118)(31,119,43)(32,44,120)(33,112,68)(34,69,105)(35,106,70)(36,71,107)(37,108,72)(38,65,109)(39,110,66)(40,67,111)(49,78,140)(50,141,79)(51,80,142)(52,143,73)(53,74,144)(54,137,75)(55,76,138)(56,139,77), (1,27,20)(2,21,28)(3,29,22)(4,23,30)(5,31,24)(6,17,32)(7,25,18)(8,19,26)(9,112,75)(10,76,105)(11,106,77)(12,78,107)(13,108,79)(14,80,109)(15,110,73)(16,74,111)(33,137,100)(34,101,138)(35,139,102)(36,103,140)(37,141,104)(38,97,142)(39,143,98)(40,99,144)(41,63,126)(42,127,64)(43,57,128)(44,121,58)(45,59,122)(46,123,60)(47,61,124)(48,125,62)(49,71,83)(50,84,72)(51,65,85)(52,86,66)(53,67,87)(54,88,68)(55,69,81)(56,82,70)(89,113,133)(90,134,114)(91,115,135)(92,136,116)(93,117,129)(94,130,118)(95,119,131)(96,132,120), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144), (1,49)(2,52)(3,55)(4,50)(5,53)(6,56)(7,51)(8,54)(9,46)(10,41)(11,44)(12,47)(13,42)(14,45)(15,48)(16,43)(17,70)(18,65)(19,68)(20,71)(21,66)(22,69)(23,72)(24,67)(25,85)(26,88)(27,83)(28,86)(29,81)(30,84)(31,87)(32,82)(33,134)(34,129)(35,132)(36,135)(37,130)(38,133)(39,136)(40,131)(57,111)(58,106)(59,109)(60,112)(61,107)(62,110)(63,105)(64,108)(73,125)(74,128)(75,123)(76,126)(77,121)(78,124)(79,127)(80,122)(89,142)(90,137)(91,140)(92,143)(93,138)(94,141)(95,144)(96,139)(97,113)(98,116)(99,119)(100,114)(101,117)(102,120)(103,115)(104,118)>;
G:=Group( (1,47,135)(2,48,136)(3,41,129)(4,42,130)(5,43,131)(6,44,132)(7,45,133)(8,46,134)(9,33,54)(10,34,55)(11,35,56)(12,36,49)(13,37,50)(14,38,51)(15,39,52)(16,40,53)(17,121,120)(18,122,113)(19,123,114)(20,124,115)(21,125,116)(22,126,117)(23,127,118)(24,128,119)(25,59,89)(26,60,90)(27,61,91)(28,62,92)(29,63,93)(30,64,94)(31,57,95)(32,58,96)(65,80,97)(66,73,98)(67,74,99)(68,75,100)(69,76,101)(70,77,102)(71,78,103)(72,79,104)(81,105,138)(82,106,139)(83,107,140)(84,108,141)(85,109,142)(86,110,143)(87,111,144)(88,112,137), (1,91,124)(2,125,92)(3,93,126)(4,127,94)(5,95,128)(6,121,96)(7,89,122)(8,123,90)(9,88,100)(10,101,81)(11,82,102)(12,103,83)(13,84,104)(14,97,85)(15,86,98)(16,99,87)(17,58,132)(18,133,59)(19,60,134)(20,135,61)(21,62,136)(22,129,63)(23,64,130)(24,131,57)(25,113,45)(26,46,114)(27,115,47)(28,48,116)(29,117,41)(30,42,118)(31,119,43)(32,44,120)(33,112,68)(34,69,105)(35,106,70)(36,71,107)(37,108,72)(38,65,109)(39,110,66)(40,67,111)(49,78,140)(50,141,79)(51,80,142)(52,143,73)(53,74,144)(54,137,75)(55,76,138)(56,139,77), (1,27,20)(2,21,28)(3,29,22)(4,23,30)(5,31,24)(6,17,32)(7,25,18)(8,19,26)(9,112,75)(10,76,105)(11,106,77)(12,78,107)(13,108,79)(14,80,109)(15,110,73)(16,74,111)(33,137,100)(34,101,138)(35,139,102)(36,103,140)(37,141,104)(38,97,142)(39,143,98)(40,99,144)(41,63,126)(42,127,64)(43,57,128)(44,121,58)(45,59,122)(46,123,60)(47,61,124)(48,125,62)(49,71,83)(50,84,72)(51,65,85)(52,86,66)(53,67,87)(54,88,68)(55,69,81)(56,82,70)(89,113,133)(90,134,114)(91,115,135)(92,136,116)(93,117,129)(94,130,118)(95,119,131)(96,132,120), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144), (1,49)(2,52)(3,55)(4,50)(5,53)(6,56)(7,51)(8,54)(9,46)(10,41)(11,44)(12,47)(13,42)(14,45)(15,48)(16,43)(17,70)(18,65)(19,68)(20,71)(21,66)(22,69)(23,72)(24,67)(25,85)(26,88)(27,83)(28,86)(29,81)(30,84)(31,87)(32,82)(33,134)(34,129)(35,132)(36,135)(37,130)(38,133)(39,136)(40,131)(57,111)(58,106)(59,109)(60,112)(61,107)(62,110)(63,105)(64,108)(73,125)(74,128)(75,123)(76,126)(77,121)(78,124)(79,127)(80,122)(89,142)(90,137)(91,140)(92,143)(93,138)(94,141)(95,144)(96,139)(97,113)(98,116)(99,119)(100,114)(101,117)(102,120)(103,115)(104,118) );
G=PermutationGroup([[(1,47,135),(2,48,136),(3,41,129),(4,42,130),(5,43,131),(6,44,132),(7,45,133),(8,46,134),(9,33,54),(10,34,55),(11,35,56),(12,36,49),(13,37,50),(14,38,51),(15,39,52),(16,40,53),(17,121,120),(18,122,113),(19,123,114),(20,124,115),(21,125,116),(22,126,117),(23,127,118),(24,128,119),(25,59,89),(26,60,90),(27,61,91),(28,62,92),(29,63,93),(30,64,94),(31,57,95),(32,58,96),(65,80,97),(66,73,98),(67,74,99),(68,75,100),(69,76,101),(70,77,102),(71,78,103),(72,79,104),(81,105,138),(82,106,139),(83,107,140),(84,108,141),(85,109,142),(86,110,143),(87,111,144),(88,112,137)], [(1,91,124),(2,125,92),(3,93,126),(4,127,94),(5,95,128),(6,121,96),(7,89,122),(8,123,90),(9,88,100),(10,101,81),(11,82,102),(12,103,83),(13,84,104),(14,97,85),(15,86,98),(16,99,87),(17,58,132),(18,133,59),(19,60,134),(20,135,61),(21,62,136),(22,129,63),(23,64,130),(24,131,57),(25,113,45),(26,46,114),(27,115,47),(28,48,116),(29,117,41),(30,42,118),(31,119,43),(32,44,120),(33,112,68),(34,69,105),(35,106,70),(36,71,107),(37,108,72),(38,65,109),(39,110,66),(40,67,111),(49,78,140),(50,141,79),(51,80,142),(52,143,73),(53,74,144),(54,137,75),(55,76,138),(56,139,77)], [(1,27,20),(2,21,28),(3,29,22),(4,23,30),(5,31,24),(6,17,32),(7,25,18),(8,19,26),(9,112,75),(10,76,105),(11,106,77),(12,78,107),(13,108,79),(14,80,109),(15,110,73),(16,74,111),(33,137,100),(34,101,138),(35,139,102),(36,103,140),(37,141,104),(38,97,142),(39,143,98),(40,99,144),(41,63,126),(42,127,64),(43,57,128),(44,121,58),(45,59,122),(46,123,60),(47,61,124),(48,125,62),(49,71,83),(50,84,72),(51,65,85),(52,86,66),(53,67,87),(54,88,68),(55,69,81),(56,82,70),(89,113,133),(90,134,114),(91,115,135),(92,136,116),(93,117,129),(94,130,118),(95,119,131),(96,132,120)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144)], [(1,49),(2,52),(3,55),(4,50),(5,53),(6,56),(7,51),(8,54),(9,46),(10,41),(11,44),(12,47),(13,42),(14,45),(15,48),(16,43),(17,70),(18,65),(19,68),(20,71),(21,66),(22,69),(23,72),(24,67),(25,85),(26,88),(27,83),(28,86),(29,81),(30,84),(31,87),(32,82),(33,134),(34,129),(35,132),(36,135),(37,130),(38,133),(39,136),(40,131),(57,111),(58,106),(59,109),(60,112),(61,107),(62,110),(63,105),(64,108),(73,125),(74,128),(75,123),(76,126),(77,121),(78,124),(79,127),(80,122),(89,142),(90,137),(91,140),(92,143),(93,138),(94,141),(95,144),(96,139),(97,113),(98,116),(99,119),(100,114),(101,117),(102,120),(103,115),(104,118)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 6A | 6B | 6C | ··· | 6N | 6O | 6P | 8A | 8B | 12A | 12B | 12C | ··· | 12AN | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 36 | 1 | 1 | 2 | ··· | 2 | 2 | 4 | 1 | 1 | 2 | ··· | 2 | 36 | 36 | 18 | 18 | 2 | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | SD16 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×SD16 | C3×C3⋊D4 | Q8⋊2S3 | C3×Q8⋊2S3 |
| kernel | C3×C32⋊11SD16 | C3×C32⋊4C8 | C3×C12⋊S3 | Q8×C33 | C32⋊11SD16 | C32⋊4C8 | C12⋊S3 | Q8×C32 | Q8×C32 | C32×C6 | C3×C12 | C33 | C3×Q8 | C3×C6 | C3×C6 | C12 | C32 | C6 | C32 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 4 | 2 | 8 | 8 | 2 | 8 | 4 | 16 | 4 | 8 |
Matrix representation of C3×C32⋊11SD16 ►in GL6(𝔽73)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 56 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 45 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 49 |
| 0 | 0 | 0 | 0 | 5 | 62 |
G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,56,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,61,13,0,0,0,0,45,0],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,5,0,0,0,0,49,62] >;
C3×C32⋊11SD16 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes_{11}{\rm SD}_{16} % in TeX
G:=Group("C3xC3^2:11SD16"); // GroupNames label
G:=SmallGroup(432,493);
// by ID
G=gap.SmallGroup(432,493);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,176,1011,514,80,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^3>;
// generators/relations