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G = C2×D8⋊3D5  order 320 = 26·5

Direct product of C2 and D8⋊3D5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D8⋊3D5, D8⋊12D10, C20.3C24, C40.32C23, Dic20⋊13C22, Dic10.1C23, (C10×D8)⋊8C2, (C2×D8)⋊13D5, C4.41(D4×D5), C10⋊2(C4○D8), (C4×D5).66D4, C20.78(C2×D4), C4.3(C23×D5), D10.21(C2×D4), (C2×C8).245D10, (C8×D5)⋊14C22, (C5×D8)⋊10C22, D4.D5⋊8C22, D4.1(C22×D5), (C5×D4).1C23, C8.38(C22×D5), (C2×Dic20)⋊19C2, (C2×D4).180D10, D4⋊2D5⋊6C22, (C2×C40).97C22, C5⋊2C8.20C23, (C22×D5).92D4, (C4×D5).60C23, C22.137(D4×D5), (C2×C20).520C23, Dic5.123(C2×D4), (C2×Dic5).283D4, C10.104(C22×D4), (D4×C10).162C22, (C2×Dic10).201C22, (D5×C2×C8)⋊5C2, C5⋊2(C2×C4○D8), C2.77(C2×D4×D5), (C2×D4.D5)⋊26C2, (C2×D4⋊2D5)⋊24C2, (C2×C10).393(C2×D4), (C2×C4×D5).326C22, (C2×C4).610(C22×D5), (C2×C5⋊2C8).291C22, SmallGroup(320,1428)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — C2×D8⋊3D5
Chief series
C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×D4⋊2D5 — C2×D8⋊3D5
Lower central
C5 — C10 — C20 — C2×D8⋊3D5

Subgroups: 926 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×12], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×4], D4 [×10], Q8 [×6], C23 [×3], D5 [×2], C10, C10 [×2], C10 [×4], C2×C8, C2×C8 [×5], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×12], Dic5 [×2], Dic5 [×4], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×8], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C5⋊2C8 [×2], C40 [×2], Dic10 [×4], Dic10 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×10], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C5×D4 [×2], C22×D5, C22×C10 [×2], C2×C4○D8, C8×D5 [×4], Dic20 [×4], C2×C5⋊2C8, D4.D5 [×8], C2×C40, C5×D8 [×4], C2×Dic10 [×2], C2×C4×D5, D4⋊2D5 [×8], D4⋊2D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10 [×2], D5×C2×C8, C2×Dic20, D8⋊3D5 [×8], C2×D4.D5 [×2], C10×D8, C2×D4⋊2D5 [×2], C2×D8⋊3D5

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C4○D8 [×2], C22×D4, C22×D5 [×7], C2×C4○D8, D4×D5 [×2], C23×D5, D8⋊3D5 [×2], C2×D4×D5, C2×D8⋊3D5

Generators and relations
 G = < a,b,c,d,e | a2=b8=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽41)

400000
040000
004000
00010
00001
,
10000
03000
001400
000400
000040
,
400000
003200
09000
000400
000040
,
10000
01000
00100
000740
000840
,
400000
040000
00100
00006
00070

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,3,0,0,0,0,0,14,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,9,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,8,0,0,0,40,40],[40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,6,0] >;
 

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E8F8G8H10A···10F10G···10N20A20B20C20D40A···40H
order12222222224444444444558888888810···1010···102020202040···40
size11114444101022555520202020222222101010102···28···844444···4

56 irreducible representations

dim111111122222222444
type++++++++++++++++-
imageC1C2C2C2C2C2C2D4D4D4D5D10D10D10C4○D8D4×D5D4×D5D8⋊3D5
kernelC2×D8⋊3D5D5×C2×C8C2×Dic20D8⋊3D5C2×D4.D5C10×D8C2×D4⋊2D5C4×D5C2×Dic5C22×D5C2×D8C2×C8D8C2×D4C10C4C22C2
# reps111821221122848228

In GAP, Magma, Sage, TeX 

C_2\times D_8\rtimes_3D_5
 
% in TeX
 
G:=Group("C2xD8:3D5");
 
// GroupNames label
 
G:=SmallGroup(320,1428);
 
// by ID
 
G=gap.SmallGroup(320,1428);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,1123,185,438,235,102,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^5=e^2=1,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,e*c*e=b^4*c,e*d*e=d^-1>;
 
// generators/relations
 

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