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## G = (C2×Q8).5F5order 320 = 26·5

### 2nd non-split extension by C2×Q8 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×Q8).5F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C2×D5⋊C8 — (C2×Q8).5F5
 Lower central C5 — C2×C10 — (C2×Q8).5F5
 Upper central C1 — C22 — C2×Q8

Generators and relations for (C2×Q8).5F5
G = < a,b,c,d,e | a2=b4=d5=1, c2=e4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe-1=ab-1, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 650 in 158 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C5⋊C8, C4×D5, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, (C22×C8)⋊C2, D5⋊C8, C4.F5, C2×C5⋊C8, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q82D5, Q8×C10, D10⋊C8, C2×D5⋊C8, C2×C4.F5, C2×Q82D5, (C2×Q8).5F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, F5, C2×C22⋊C4, C8○D4, C2×F5, (C22×C8)⋊C2, C22⋊F5, C22×F5, Q8.F5, C2×C22⋊F5, (C2×Q8).5F5

Smallest permutation representation of (C2×Q8).5F5
On 160 points
Generators in S160
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 76)(10 77)(11 78)(12 79)(13 80)(14 73)(15 74)(16 75)(17 115)(18 116)(19 117)(20 118)(21 119)(22 120)(23 113)(24 114)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 158)(34 159)(35 160)(36 153)(37 154)(38 155)(39 156)(40 157)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 97)(56 98)(65 108)(66 109)(67 110)(68 111)(69 112)(70 105)(71 106)(72 107)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)(129 152)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)
(1 102 5 98)(2 49 6 53)(3 104 7 100)(4 51 8 55)(9 122 13 126)(10 31 14 27)(11 124 15 128)(12 25 16 29)(17 131 21 135)(18 151 22 147)(19 133 23 129)(20 145 24 149)(26 80 30 76)(28 74 32 78)(33 65 37 69)(34 105 38 109)(35 67 39 71)(36 107 40 111)(41 83 45 87)(42 89 46 93)(43 85 47 81)(44 91 48 95)(50 63 54 59)(52 57 56 61)(58 103 62 99)(60 97 64 101)(66 159 70 155)(68 153 72 157)(73 123 77 127)(75 125 79 121)(82 144 86 140)(84 138 88 142)(90 139 94 143)(92 141 96 137)(106 160 110 156)(108 154 112 158)(113 152 117 148)(114 134 118 130)(115 146 119 150)(116 136 120 132)
(1 104 5 100)(2 97 6 101)(3 98 7 102)(4 99 8 103)(9 108 13 112)(10 109 14 105)(11 110 15 106)(12 111 16 107)(17 48 21 44)(18 41 22 45)(19 42 23 46)(20 43 24 47)(25 40 29 36)(26 33 30 37)(27 34 31 38)(28 35 32 39)(49 60 53 64)(50 61 54 57)(51 62 55 58)(52 63 56 59)(65 80 69 76)(66 73 70 77)(67 74 71 78)(68 75 72 79)(81 149 85 145)(82 150 86 146)(83 151 87 147)(84 152 88 148)(89 133 93 129)(90 134 94 130)(91 135 95 131)(92 136 96 132)(113 142 117 138)(114 143 118 139)(115 144 119 140)(116 137 120 141)(121 157 125 153)(122 158 126 154)(123 159 127 155)(124 160 128 156)
(1 12 132 143 38)(2 144 13 39 133)(3 40 137 134 14)(4 135 33 15 138)(5 16 136 139 34)(6 140 9 35 129)(7 36 141 130 10)(8 131 37 11 142)(17 65 128 88 51)(18 81 66 52 121)(19 53 82 122 67)(20 123 54 68 83)(21 69 124 84 55)(22 85 70 56 125)(23 49 86 126 71)(24 127 50 72 87)(25 116 90 109 102)(26 110 117 103 91)(27 104 111 92 118)(28 93 97 119 112)(29 120 94 105 98)(30 106 113 99 95)(31 100 107 96 114)(32 89 101 115 108)(41 149 73 63 157)(42 64 150 158 74)(43 159 57 75 151)(44 76 160 152 58)(45 145 77 59 153)(46 60 146 154 78)(47 155 61 79 147)(48 80 156 148 62)
(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)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5 8A ··· 8H 8I 8J 8K 8L 10A 10B 10C 20A ··· 20F order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 8 ··· 8 8 8 8 8 10 10 10 20 ··· 20 size 1 1 1 1 10 10 20 20 2 2 4 4 5 5 5 5 4 10 ··· 10 20 20 20 20 4 4 4 8 ··· 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 4 8 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 D4 C8○D4 F5 C2×F5 C22⋊F5 Q8.F5 kernel (C2×Q8).5F5 D10⋊C8 C2×D5⋊C8 C2×C4.F5 C2×Q8⋊2D5 C2×D20 Q8×C10 C4×D5 C10 C2×Q8 C2×C4 C4 C2 # reps 1 4 1 1 1 6 2 4 8 1 3 4 2

Matrix representation of (C2×Q8).5F5 in GL8(𝔽41)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 40 40 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 35 39 0 0 0 0 0 0 38 6 0 0 0 0 0 0 0 0 35 0 16 16 0 0 0 0 16 16 0 35 0 0 0 0 25 19 25 0 0 0 0 0 6 22 22 6

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,35,38,0,0,0,0,0,0,39,6,0,0,0,0,0,0,0,0,35,16,25,6,0,0,0,0,0,16,19,22,0,0,0,0,16,0,25,22,0,0,0,0,16,35,0,6] >;

(C2×Q8).5F5 in GAP, Magma, Sage, TeX

(C_2\times Q_8)._5F_5
% in TeX

G:=Group("(C2xQ8).5F5");
// GroupNames label

G:=SmallGroup(320,1125);
// by ID

G=gap.SmallGroup(320,1125);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^5=1,c^2=e^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e^-1=a*b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

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