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G = C10.1072- 1+4order 320 = 26·5

62nd non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.1072- 1+4, (C5×Q8)⋊18D4, Q89(C5⋊D4), C57(Q85D4), C202D442C2, (Q8×Dic5)⋊29C2, C20.267(C2×D4), D1016(C4○D4), (C2×D4).236D10, (C2×Q8).210D10, C20.48D440C2, C20.17D431C2, (C2×C20).561C23, (C2×C10).317C24, C10.167(C22×D4), (C22×C4).287D10, (D4×C10).276C22, C4⋊Dic5.260C22, (Q8×C10).243C22, C22.326(C23×D5), C23.138(C22×D5), C23.D5.76C22, (C22×C20).296C22, (C22×C10).243C23, (C2×Dic5).164C23, (C4×Dic5).184C22, (C22×D5).254C23, C2.70(D4.10D10), D10⋊C4.164C22, (C2×Dic10).210C22, C10.D4.101C22, (C2×Q8×D5)⋊19C2, (C2×C4○D4)⋊9D5, (C10×C4○D4)⋊9C2, (C4×C5⋊D4)⋊30C2, C4.73(C2×C5⋊D4), C2.105(D5×C4○D4), C10.217(C2×C4○D4), (C2×C4×D5).178C22, C2.40(C22×C5⋊D4), (C2×C4).640(C22×D5), (C2×C5⋊D4).149C22, SmallGroup(320,1503)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.1072- 1+4
C1C5C10C2×C10C22×D5C2×C4×D5C2×Q8×D5 — C10.1072- 1+4
C5C2×C10 — C10.1072- 1+4
C1C22C2×C4○D4

Generators and relations for C10.1072- 1+4
 G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=a5b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, dbd-1=a5b, be=eb, dcd-1=a5c, ce=ec, ede-1=a5b2d >

Subgroups: 902 in 290 conjugacy classes, 113 normal (22 characteristic)
C1, C2 [×3], C2 [×5], C4 [×6], C4 [×8], C22, C22 [×13], C5, C2×C4, C2×C4 [×3], C2×C4 [×19], D4 [×12], Q8 [×4], Q8 [×6], C23 [×3], C23, D5 [×2], C10 [×3], C10 [×3], C42 [×3], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic5 [×7], C20 [×6], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×9], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×6], C4×D5 [×6], C2×Dic5, C2×Dic5 [×6], C5⋊D4 [×6], C2×C20, C2×C20 [×3], C2×C20 [×6], C5×D4 [×6], C5×Q8 [×4], C22×D5, C22×C10 [×3], Q85D4, C4×Dic5 [×3], C10.D4 [×3], C4⋊Dic5 [×3], D10⋊C4, C23.D5 [×9], C2×Dic10 [×3], C2×C4×D5 [×3], Q8×D5 [×4], C2×C5⋊D4 [×3], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×4], C20.48D4 [×3], C4×C5⋊D4 [×3], C20.17D4 [×3], C202D4 [×3], Q8×Dic5, C2×Q8×D5, C10×C4○D4, C10.1072- 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- 1+4, C5⋊D4 [×4], C22×D5 [×7], Q85D4, C2×C5⋊D4 [×6], C23×D5, D5×C4○D4, D4.10D10, C22×C5⋊D4, C10.1072- 1+4

Smallest permutation representation of C10.1072- 1+4
On 160 points
Generators in S160
(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)
(1 125 28 113)(2 124 29 112)(3 123 30 111)(4 122 21 120)(5 121 22 119)(6 130 23 118)(7 129 24 117)(8 128 25 116)(9 127 26 115)(10 126 27 114)(11 53 153 65)(12 52 154 64)(13 51 155 63)(14 60 156 62)(15 59 157 61)(16 58 158 70)(17 57 159 69)(18 56 160 68)(19 55 151 67)(20 54 152 66)(31 100 43 102)(32 99 44 101)(33 98 45 110)(34 97 46 109)(35 96 47 108)(36 95 48 107)(37 94 49 106)(38 93 50 105)(39 92 41 104)(40 91 42 103)(71 140 83 142)(72 139 84 141)(73 138 85 150)(74 137 86 149)(75 136 87 148)(76 135 88 147)(77 134 89 146)(78 133 90 145)(79 132 81 144)(80 131 82 143)
(1 118)(2 119)(3 120)(4 111)(5 112)(6 113)(7 114)(8 115)(9 116)(10 117)(11 58)(12 59)(13 60)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 121)(30 122)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(61 154)(62 155)(63 156)(64 157)(65 158)(66 159)(67 160)(68 151)(69 152)(70 153)(71 131)(72 132)(73 133)(74 134)(75 135)(76 136)(77 137)(78 138)(79 139)(80 140)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)
(1 65 23 58)(2 64 24 57)(3 63 25 56)(4 62 26 55)(5 61 27 54)(6 70 28 53)(7 69 29 52)(8 68 30 51)(9 67 21 60)(10 66 22 59)(11 113 158 130)(12 112 159 129)(13 111 160 128)(14 120 151 127)(15 119 152 126)(16 118 153 125)(17 117 154 124)(18 116 155 123)(19 115 156 122)(20 114 157 121)(31 87 48 80)(32 86 49 79)(33 85 50 78)(34 84 41 77)(35 83 42 76)(36 82 43 75)(37 81 44 74)(38 90 45 73)(39 89 46 72)(40 88 47 71)(91 142 108 135)(92 141 109 134)(93 150 110 133)(94 149 101 132)(95 148 102 131)(96 147 103 140)(97 146 104 139)(98 145 105 138)(99 144 106 137)(100 143 107 136)
(1 98 23 105)(2 99 24 106)(3 100 25 107)(4 91 26 108)(5 92 27 109)(6 93 28 110)(7 94 29 101)(8 95 30 102)(9 96 21 103)(10 97 22 104)(11 85 158 78)(12 86 159 79)(13 87 160 80)(14 88 151 71)(15 89 152 72)(16 90 153 73)(17 81 154 74)(18 82 155 75)(19 83 156 76)(20 84 157 77)(31 128 48 111)(32 129 49 112)(33 130 50 113)(34 121 41 114)(35 122 42 115)(36 123 43 116)(37 124 44 117)(38 125 45 118)(39 126 46 119)(40 127 47 120)(51 148 68 131)(52 149 69 132)(53 150 70 133)(54 141 61 134)(55 142 62 135)(56 143 63 136)(57 144 64 137)(58 145 65 138)(59 146 66 139)(60 147 67 140)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4H4I4J4K···4P5A5B10A···10F10G···10R20A···20H20I···20T
order1222222224···4444···45510···1010···1020···2020···20
size111144410102···2101020···20222···24···42···24···4

65 irreducible representations

dim111111112222222444
type+++++++++++++--
imageC1C2C2C2C2C2C2C2D4D5C4○D4D10D10D10C5⋊D42- 1+4D5×C4○D4D4.10D10
kernelC10.1072- 1+4C20.48D4C4×C5⋊D4C20.17D4C202D4Q8×Dic5C2×Q8×D5C10×C4○D4C5×Q8C2×C4○D4D10C22×C4C2×D4C2×Q8Q8C10C2C2
# reps1333311142466216144

Matrix representation of C10.1072- 1+4 in GL6(𝔽41)

35350000
6400000
0040000
0004000
0000400
0000040
,
100000
6400000
00122300
00242900
00002815
00003813
,
100000
010000
00291800
0081200
00001326
0000328
,
4000000
3510000
001000
00154000
000094
0000032
,
4000000
0400000
0040000
0004000
00003529
0000276

G:=sub<GL(6,GF(41))| [35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,12,24,0,0,0,0,23,29,0,0,0,0,0,0,28,38,0,0,0,0,15,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,8,0,0,0,0,18,12,0,0,0,0,0,0,13,3,0,0,0,0,26,28],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,15,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,4,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,27,0,0,0,0,29,6] >;

C10.1072- 1+4 in GAP, Magma, Sage, TeX

C_{10}._{107}2_-^{1+4}
% in TeX

G:=Group("C10.107ES-(2,2)");
// GroupNames label

G:=SmallGroup(320,1503);
// by ID

G=gap.SmallGroup(320,1503);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,136,12550]);
// Polycyclic

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

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