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G = D4⋊D56C4order 320 = 26·5

2nd semidirect product of D4⋊D5 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4⋊D56C4, D42(C4×D5), D2013(C2×C4), C53(D8⋊C4), (D4×Dic5)⋊2C2, C408C418C2, C10.63(C4×D4), D4⋊C420D5, D208C42C2, C4⋊C4.139D10, (C2×C8).171D10, C20.Q88C2, D205C421C2, C22.72(D4×D5), (C2×D4).138D10, C2.4(D8⋊D5), C2.2(D40⋊C2), C20.43(C22×C4), C20.153(C4○D4), C4.50(D42D5), C10.57(C8⋊C22), (C2×C20).225C23, (C2×C40).188C22, (C2×Dic5).201D4, (D4×C10).46C22, (C2×D20).57C22, C4⋊Dic5.77C22, (C4×Dic5).18C22, C2.17(Dic54D4), C4.8(C2×C4×D5), C52C82(C2×C4), (C5×D4)⋊14(C2×C4), (C2×D4⋊D5).2C2, (C5×D4⋊C4)⋊26C2, (C2×C10).238(C2×D4), (C5×C4⋊C4).26C22, (C2×C52C8).23C22, (C2×C4).332(C22×D5), SmallGroup(320,412)

Series: Derived Chief Lower central Upper central

C1C20 — D4⋊D56C4
C1C5C10C2×C10C2×C20C2×D20C2×D4⋊D5 — D4⋊D56C4
C5C10C20 — D4⋊D56C4
C1C22C2×C4D4⋊C4

Generators and relations for D4⋊D56C4
 G = < a,b,c,d,e | a4=b2=c5=d2=e4=1, bab=dad=eae-1=a-1, ac=ca, bc=cb, dbd=ab, ebe-1=a-1b, dcd=c-1, ce=ec, de=ed >

Subgroups: 566 in 132 conjugacy classes, 49 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×3], C2×C4, C2×C4 [×8], D4 [×2], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×4], C22×C4 [×2], C2×D4, C2×D4, Dic5 [×3], C20 [×2], C20, D10 [×4], C2×C10, C2×C10 [×4], C8⋊C4, D4⋊C4, D4⋊C4, C4.Q8, C4×D4 [×2], C2×D8, C52C8 [×2], C40, C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5 [×3], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×D5, C22×C10, D8⋊C4, C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4, D4⋊D5 [×4], C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C22×Dic5, D4×C10, C20.Q8, C408C4, D205C4, C5×D4⋊C4, D208C4, C2×D4⋊D5, D4×Dic5, D4⋊D56C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8⋊C22 [×2], C4×D5 [×2], C22×D5, D8⋊C4, C2×C4×D5, D4×D5, D42D5, Dic54D4, D8⋊D5, D40⋊C2, D4⋊D56C4

Smallest permutation representation of D4⋊D56C4
On 160 points
Generators in S160
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 136 126 131)(122 137 127 132)(123 138 128 133)(124 139 129 134)(125 140 130 135)(141 156 146 151)(142 157 147 152)(143 158 148 153)(144 159 149 154)(145 160 150 155)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 111)(7 112)(8 113)(9 114)(10 115)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 151)(42 152)(43 153)(44 154)(45 155)(46 156)(47 157)(48 158)(49 159)(50 160)(51 146)(52 147)(53 148)(54 149)(55 150)(56 141)(57 142)(58 143)(59 144)(60 145)(61 131)(62 132)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 126)(72 127)(73 128)(74 129)(75 130)(76 121)(77 122)(78 123)(79 124)(80 125)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 30)(7 29)(8 28)(9 27)(10 26)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(41 65)(42 64)(43 63)(44 62)(45 61)(46 70)(47 69)(48 68)(49 67)(50 66)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(81 115)(82 114)(83 113)(84 112)(85 111)(86 120)(87 119)(88 118)(89 117)(90 116)(91 105)(92 104)(93 103)(94 102)(95 101)(96 110)(97 109)(98 108)(99 107)(100 106)(121 160)(122 159)(123 158)(124 157)(125 156)(126 155)(127 154)(128 153)(129 152)(130 151)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 143)(139 142)(140 141)
(1 61 21 41)(2 62 22 42)(3 63 23 43)(4 64 24 44)(5 65 25 45)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 151 101 131)(82 152 102 132)(83 153 103 133)(84 154 104 134)(85 155 105 135)(86 156 106 136)(87 157 107 137)(88 158 108 138)(89 159 109 139)(90 160 110 140)(91 146 111 126)(92 147 112 127)(93 148 113 128)(94 149 114 129)(95 150 115 130)(96 141 116 121)(97 142 117 122)(98 143 118 123)(99 144 119 124)(100 145 120 125)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222222444444444455888810···1010101010202020202020202040···40
size11114420202244101010102020224420202···28888444488884···4

50 irreducible representations

dim111111111222222244444
type++++++++++++++-++
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10D10D10C4×D5C8⋊C22D42D5D4×D5D8⋊D5D40⋊C2
kernelD4⋊D56C4C20.Q8C408C4D205C4C5×D4⋊C4D208C4C2×D4⋊D5D4×Dic5D4⋊D5C2×Dic5D4⋊C4C20C4⋊C4C2×C8C2×D4D4C10C4C22C2C2
# reps111111118222222822244

Matrix representation of D4⋊D56C4 in GL6(𝔽41)

100000
010000
000010
000001
0040000
0004000
,
100000
010000
003353335
0068638
003335386
006383533
,
7400000
8400000
0064000
001000
0000640
000010
,
0350000
3400000
0064000
00353500
0000351
000066
,
900000
090000
00240213
000242839
00213170
002839017

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,6,33,6,0,0,35,8,35,38,0,0,33,6,38,35,0,0,35,38,6,33],[7,8,0,0,0,0,40,40,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,6,1,0,0,0,0,40,0],[0,34,0,0,0,0,35,0,0,0,0,0,0,0,6,35,0,0,0,0,40,35,0,0,0,0,0,0,35,6,0,0,0,0,1,6],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,24,0,2,28,0,0,0,24,13,39,0,0,2,28,17,0,0,0,13,39,0,17] >;

D4⋊D56C4 in GAP, Magma, Sage, TeX

D_4\rtimes D_5\rtimes_6C_4
% in TeX

G:=Group("D4:D5:6C4");
// GroupNames label

G:=SmallGroup(320,412);
// by ID

G=gap.SmallGroup(320,412);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,219,58,1684,851,438,102,12550]);
// Polycyclic

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

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