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## G = C2×D4×Dic5order 320 = 26·5

### Direct product of C2, D4 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D4×Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C2×D4×Dic5
 Lower central C5 — C10 — C2×D4×Dic5
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×D4×Dic5
G = < a,b,c,d,e | a2=b4=c2=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 1102 in 426 conjugacy classes, 215 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×10], C22, C22 [×14], C22 [×24], C5, C2×C4 [×6], C2×C4 [×34], D4 [×16], C23, C23 [×12], C23 [×8], C10 [×3], C10 [×4], C10 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×20], C2×D4 [×12], C24 [×2], Dic5 [×4], Dic5 [×6], C20 [×4], C2×C10, C2×C10 [×14], C2×C10 [×24], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C2×Dic5 [×12], C2×Dic5 [×22], C2×C20 [×6], C5×D4 [×16], C22×C10, C22×C10 [×12], C22×C10 [×8], C2×C4×D4, C4×Dic5 [×4], C4⋊Dic5 [×4], C23.D5 [×8], C22×Dic5 [×2], C22×Dic5 [×10], C22×Dic5 [×8], C22×C20, D4×C10 [×12], C23×C10 [×2], C2×C4×Dic5, C2×C4⋊Dic5, D4×Dic5 [×8], C2×C23.D5 [×2], C23×Dic5 [×2], D4×C2×C10, C2×D4×Dic5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D5, C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, Dic5 [×8], D10 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×Dic5 [×28], C22×D5 [×7], C2×C4×D4, D4×D5 [×2], D42D5 [×2], C22×Dic5 [×14], C23×D5, D4×Dic5 [×4], C2×D4×D5, C2×D42D5, C23×Dic5, C2×D4×Dic5

Smallest permutation representation of C2×D4×Dic5
On 160 points
Generators in S160
(1 68)(2 69)(3 70)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 122)(12 123)(13 124)(14 125)(15 126)(16 127)(17 128)(18 129)(19 130)(20 121)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 88)(42 89)(43 90)(44 81)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(91 146)(92 147)(93 148)(94 149)(95 150)(96 141)(97 142)(98 143)(99 144)(100 145)(101 136)(102 137)(103 138)(104 139)(105 140)(106 131)(107 132)(108 133)(109 134)(110 135)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(1 85 28 78)(2 86 29 79)(3 87 30 80)(4 88 21 71)(5 89 22 72)(6 90 23 73)(7 81 24 74)(8 82 25 75)(9 83 26 76)(10 84 27 77)(11 102 157 97)(12 103 158 98)(13 104 159 99)(14 105 160 100)(15 106 151 91)(16 107 152 92)(17 108 153 93)(18 109 154 94)(19 110 155 95)(20 101 156 96)(31 61 41 51)(32 62 42 52)(33 63 43 53)(34 64 44 54)(35 65 45 55)(36 66 46 56)(37 67 47 57)(38 68 48 58)(39 69 49 59)(40 70 50 60)(111 146 126 131)(112 147 127 132)(113 148 128 133)(114 149 129 134)(115 150 130 135)(116 141 121 136)(117 142 122 137)(118 143 123 138)(119 144 124 139)(120 145 125 140)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 51)(10 52)(11 127)(12 128)(13 129)(14 130)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 66)(22 67)(23 68)(24 69)(25 70)(26 61)(27 62)(28 63)(29 64)(30 65)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 81)(50 82)(91 136)(92 137)(93 138)(94 139)(95 140)(96 131)(97 132)(98 133)(99 134)(100 135)(101 146)(102 147)(103 148)(104 149)(105 150)(106 141)(107 142)(108 143)(109 144)(110 145)(111 156)(112 157)(113 158)(114 159)(115 160)(116 151)(117 152)(118 153)(119 154)(120 155)
(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)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 137 6 132)(2 136 7 131)(3 135 8 140)(4 134 9 139)(5 133 10 138)(11 33 16 38)(12 32 17 37)(13 31 18 36)(14 40 19 35)(15 39 20 34)(21 149 26 144)(22 148 27 143)(23 147 28 142)(24 146 29 141)(25 145 30 150)(41 154 46 159)(42 153 47 158)(43 152 48 157)(44 151 49 156)(45 160 50 155)(51 94 56 99)(52 93 57 98)(53 92 58 97)(54 91 59 96)(55 100 60 95)(61 109 66 104)(62 108 67 103)(63 107 68 102)(64 106 69 101)(65 105 70 110)(71 129 76 124)(72 128 77 123)(73 127 78 122)(74 126 79 121)(75 125 80 130)(81 111 86 116)(82 120 87 115)(83 119 88 114)(84 118 89 113)(85 117 90 112)

G:=sub<Sym(160)| (1,68)(2,69)(3,70)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,122)(12,123)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,121)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,88)(42,89)(43,90)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(91,146)(92,147)(93,148)(94,149)(95,150)(96,141)(97,142)(98,143)(99,144)(100,145)(101,136)(102,137)(103,138)(104,139)(105,140)(106,131)(107,132)(108,133)(109,134)(110,135)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,85,28,78)(2,86,29,79)(3,87,30,80)(4,88,21,71)(5,89,22,72)(6,90,23,73)(7,81,24,74)(8,82,25,75)(9,83,26,76)(10,84,27,77)(11,102,157,97)(12,103,158,98)(13,104,159,99)(14,105,160,100)(15,106,151,91)(16,107,152,92)(17,108,153,93)(18,109,154,94)(19,110,155,95)(20,101,156,96)(31,61,41,51)(32,62,42,52)(33,63,43,53)(34,64,44,54)(35,65,45,55)(36,66,46,56)(37,67,47,57)(38,68,48,58)(39,69,49,59)(40,70,50,60)(111,146,126,131)(112,147,127,132)(113,148,128,133)(114,149,129,134)(115,150,130,135)(116,141,121,136)(117,142,122,137)(118,143,123,138)(119,144,124,139)(120,145,125,140), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,51)(10,52)(11,127)(12,128)(13,129)(14,130)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,81)(50,82)(91,136)(92,137)(93,138)(94,139)(95,140)(96,131)(97,132)(98,133)(99,134)(100,135)(101,146)(102,147)(103,148)(104,149)(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)(112,157)(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)(120,155), (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)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,137,6,132)(2,136,7,131)(3,135,8,140)(4,134,9,139)(5,133,10,138)(11,33,16,38)(12,32,17,37)(13,31,18,36)(14,40,19,35)(15,39,20,34)(21,149,26,144)(22,148,27,143)(23,147,28,142)(24,146,29,141)(25,145,30,150)(41,154,46,159)(42,153,47,158)(43,152,48,157)(44,151,49,156)(45,160,50,155)(51,94,56,99)(52,93,57,98)(53,92,58,97)(54,91,59,96)(55,100,60,95)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,129,76,124)(72,128,77,123)(73,127,78,122)(74,126,79,121)(75,125,80,130)(81,111,86,116)(82,120,87,115)(83,119,88,114)(84,118,89,113)(85,117,90,112)>;

G:=Group( (1,68)(2,69)(3,70)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,122)(12,123)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,121)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,88)(42,89)(43,90)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(91,146)(92,147)(93,148)(94,149)(95,150)(96,141)(97,142)(98,143)(99,144)(100,145)(101,136)(102,137)(103,138)(104,139)(105,140)(106,131)(107,132)(108,133)(109,134)(110,135)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,85,28,78)(2,86,29,79)(3,87,30,80)(4,88,21,71)(5,89,22,72)(6,90,23,73)(7,81,24,74)(8,82,25,75)(9,83,26,76)(10,84,27,77)(11,102,157,97)(12,103,158,98)(13,104,159,99)(14,105,160,100)(15,106,151,91)(16,107,152,92)(17,108,153,93)(18,109,154,94)(19,110,155,95)(20,101,156,96)(31,61,41,51)(32,62,42,52)(33,63,43,53)(34,64,44,54)(35,65,45,55)(36,66,46,56)(37,67,47,57)(38,68,48,58)(39,69,49,59)(40,70,50,60)(111,146,126,131)(112,147,127,132)(113,148,128,133)(114,149,129,134)(115,150,130,135)(116,141,121,136)(117,142,122,137)(118,143,123,138)(119,144,124,139)(120,145,125,140), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,51)(10,52)(11,127)(12,128)(13,129)(14,130)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,66)(22,67)(23,68)(24,69)(25,70)(26,61)(27,62)(28,63)(29,64)(30,65)(31,76)(32,77)(33,78)(34,79)(35,80)(36,71)(37,72)(38,73)(39,74)(40,75)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,81)(50,82)(91,136)(92,137)(93,138)(94,139)(95,140)(96,131)(97,132)(98,133)(99,134)(100,135)(101,146)(102,147)(103,148)(104,149)(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)(112,157)(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)(120,155), (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)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,137,6,132)(2,136,7,131)(3,135,8,140)(4,134,9,139)(5,133,10,138)(11,33,16,38)(12,32,17,37)(13,31,18,36)(14,40,19,35)(15,39,20,34)(21,149,26,144)(22,148,27,143)(23,147,28,142)(24,146,29,141)(25,145,30,150)(41,154,46,159)(42,153,47,158)(43,152,48,157)(44,151,49,156)(45,160,50,155)(51,94,56,99)(52,93,57,98)(53,92,58,97)(54,91,59,96)(55,100,60,95)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,129,76,124)(72,128,77,123)(73,127,78,122)(74,126,79,121)(75,125,80,130)(81,111,86,116)(82,120,87,115)(83,119,88,114)(84,118,89,113)(85,117,90,112) );

G=PermutationGroup([(1,68),(2,69),(3,70),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,122),(12,123),(13,124),(14,125),(15,126),(16,127),(17,128),(18,129),(19,130),(20,121),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,88),(42,89),(43,90),(44,81),(45,82),(46,83),(47,84),(48,85),(49,86),(50,87),(91,146),(92,147),(93,148),(94,149),(95,150),(96,141),(97,142),(98,143),(99,144),(100,145),(101,136),(102,137),(103,138),(104,139),(105,140),(106,131),(107,132),(108,133),(109,134),(110,135),(111,151),(112,152),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160)], [(1,85,28,78),(2,86,29,79),(3,87,30,80),(4,88,21,71),(5,89,22,72),(6,90,23,73),(7,81,24,74),(8,82,25,75),(9,83,26,76),(10,84,27,77),(11,102,157,97),(12,103,158,98),(13,104,159,99),(14,105,160,100),(15,106,151,91),(16,107,152,92),(17,108,153,93),(18,109,154,94),(19,110,155,95),(20,101,156,96),(31,61,41,51),(32,62,42,52),(33,63,43,53),(34,64,44,54),(35,65,45,55),(36,66,46,56),(37,67,47,57),(38,68,48,58),(39,69,49,59),(40,70,50,60),(111,146,126,131),(112,147,127,132),(113,148,128,133),(114,149,129,134),(115,150,130,135),(116,141,121,136),(117,142,122,137),(118,143,123,138),(119,144,124,139),(120,145,125,140)], [(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,51),(10,52),(11,127),(12,128),(13,129),(14,130),(15,121),(16,122),(17,123),(18,124),(19,125),(20,126),(21,66),(22,67),(23,68),(24,69),(25,70),(26,61),(27,62),(28,63),(29,64),(30,65),(31,76),(32,77),(33,78),(34,79),(35,80),(36,71),(37,72),(38,73),(39,74),(40,75),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,81),(50,82),(91,136),(92,137),(93,138),(94,139),(95,140),(96,131),(97,132),(98,133),(99,134),(100,135),(101,146),(102,147),(103,148),(104,149),(105,150),(106,141),(107,142),(108,143),(109,144),(110,145),(111,156),(112,157),(113,158),(114,159),(115,160),(116,151),(117,152),(118,153),(119,154),(120,155)], [(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),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,137,6,132),(2,136,7,131),(3,135,8,140),(4,134,9,139),(5,133,10,138),(11,33,16,38),(12,32,17,37),(13,31,18,36),(14,40,19,35),(15,39,20,34),(21,149,26,144),(22,148,27,143),(23,147,28,142),(24,146,29,141),(25,145,30,150),(41,154,46,159),(42,153,47,158),(43,152,48,157),(44,151,49,156),(45,160,50,155),(51,94,56,99),(52,93,57,98),(53,92,58,97),(54,91,59,96),(55,100,60,95),(61,109,66,104),(62,108,67,103),(63,107,68,102),(64,106,69,101),(65,105,70,110),(71,129,76,124),(72,128,77,123),(73,127,78,122),(74,126,79,121),(75,125,80,130),(81,111,86,116),(82,120,87,115),(83,119,88,114),(84,118,89,113),(85,117,90,112)])

80 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A 4B 4C 4D 4E ··· 4L 4M ··· 4X 5A 5B 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 ··· 2 2 ··· 2 4 4 4 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 2 2 2 2 5 ··· 5 10 ··· 10 2 2 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 C4 D4 D5 C4○D4 D10 Dic5 D10 D10 D4×D5 D4⋊2D5 kernel C2×D4×Dic5 C2×C4×Dic5 C2×C4⋊Dic5 D4×Dic5 C2×C23.D5 C23×Dic5 D4×C2×C10 D4×C10 C2×Dic5 C22×D4 C2×C10 C22×C4 C2×D4 C2×D4 C24 C22 C22 # reps 1 1 1 8 2 2 1 16 4 2 4 2 16 8 4 4 4

Matrix representation of C2×D4×Dic5 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 37 0 0 0 0 21 1
,
 40 0 0 0 0 0 0 40 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 20 40
,
 23 0 0 0 0 0 0 25 0 0 0 0 0 0 0 40 0 0 0 0 1 35 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 4 31 0 0 0 0 14 37 0 0 0 0 0 0 32 0 0 0 0 0 0 32

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,21,0,0,0,0,37,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,20,0,0,0,0,0,40],[23,0,0,0,0,0,0,25,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,4,14,0,0,0,0,31,37,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;

C2×D4×Dic5 in GAP, Magma, Sage, TeX

C_2\times D_4\times {\rm Dic}_5
% in TeX

G:=Group("C2xD4xDic5");
// GroupNames label

G:=SmallGroup(320,1467);
// by ID

G=gap.SmallGroup(320,1467);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,297,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^10=1,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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