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## G = C2×C20.C23order 320 = 26·5

### Direct product of C2 and C20.C23

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C20.C23
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C4○D20 — C2×C20.C23
 Lower central C5 — C10 — C20 — C2×C20.C23
 Upper central C1 — C22 — C22×C4 — C22×Q8

Generators and relations for C2×C20.C23
G = < a,b,c,d,e | a2=b20=c2=1, d2=e2=b10, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b11, cd=dc, ece-1=b5c, ede-1=b10d >

Subgroups: 798 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×4], Q8 [×9], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×2], C20 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C52C8 [×4], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×6], C5×Q8 [×4], C5×Q8 [×6], C22×D5, C22×C10, C2×C8.C22, C2×C52C8 [×2], C4.Dic5 [×4], Q8⋊D5 [×8], C5⋊Q16 [×8], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C22×C20, Q8×C10 [×6], Q8×C10 [×3], C2×C4.Dic5, C2×Q8⋊D5 [×2], C20.C23 [×8], C2×C5⋊Q16 [×2], C2×C4○D20, Q8×C2×C10, C2×C20.C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8.C22 [×2], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C8.C22, C2×C5⋊D4 [×6], C23×D5, C20.C23 [×2], C22×C5⋊D4, C2×C20.C23

Smallest permutation representation of C2×C20.C23
On 160 points
Generators in S160
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 101)(37 102)(38 103)(39 104)(40 105)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 127)(60 128)(61 152)(62 153)(63 154)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)
(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)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 83)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(101 106)(102 105)(103 104)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(134 140)(135 139)(136 138)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(153 160)(154 159)(155 158)(156 157)
(1 137 11 127)(2 138 12 128)(3 139 13 129)(4 140 14 130)(5 121 15 131)(6 122 16 132)(7 123 17 133)(8 124 18 134)(9 125 19 135)(10 126 20 136)(21 159 31 149)(22 160 32 150)(23 141 33 151)(24 142 34 152)(25 143 35 153)(26 144 36 154)(27 145 37 155)(28 146 38 156)(29 147 39 157)(30 148 40 158)(41 94 51 84)(42 95 52 85)(43 96 53 86)(44 97 54 87)(45 98 55 88)(46 99 56 89)(47 100 57 90)(48 81 58 91)(49 82 59 92)(50 83 60 93)(61 109 71 119)(62 110 72 120)(63 111 73 101)(64 112 74 102)(65 113 75 103)(66 114 76 104)(67 115 77 105)(68 116 78 106)(69 117 79 107)(70 118 80 108)
(1 26 11 36)(2 37 12 27)(3 28 13 38)(4 39 14 29)(5 30 15 40)(6 21 16 31)(7 32 17 22)(8 23 18 33)(9 34 19 24)(10 25 20 35)(41 75 51 65)(42 66 52 76)(43 77 53 67)(44 68 54 78)(45 79 55 69)(46 70 56 80)(47 61 57 71)(48 72 58 62)(49 63 59 73)(50 74 60 64)(81 110 91 120)(82 101 92 111)(83 112 93 102)(84 103 94 113)(85 114 95 104)(86 105 96 115)(87 116 97 106)(88 107 98 117)(89 118 99 108)(90 109 100 119)(121 158 131 148)(122 149 132 159)(123 160 133 150)(124 151 134 141)(125 142 135 152)(126 153 136 143)(127 144 137 154)(128 155 138 145)(129 146 139 156)(130 157 140 147)

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 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 D4 D4 D5 D10 D10 C5⋊D4 C5⋊D4 C8.C22 C20.C23 kernel C2×C20.C23 C2×C4.Dic5 C2×Q8⋊D5 C20.C23 C2×C5⋊Q16 C2×C4○D20 Q8×C2×C10 C2×C20 C22×C10 C22×Q8 C22×C4 C2×Q8 C2×C4 C23 C10 C2 # reps 1 1 2 8 2 1 1 3 1 2 2 12 12 4 2 8

Matrix representation of C2×C20.C23 in GL6(𝔽41)

 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 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 6 1 0 0 0 0 40 0 0 0 35 40 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 40 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 6 0 0 0 0 35 23 0 0 23 35 0 0 0 0 6 18 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 31 36 20 5 0 0 5 20 36 31 0 0 20 5 10 5 0 0 36 31 36 21

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,35,1,0,0,0,0,40,0,0,0,6,40,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,35,1,0,0,0,0,6,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,23,6,0,0,0,0,35,18,0,0,18,35,0,0,0,0,6,23,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,31,5,20,36,0,0,36,20,5,31,0,0,20,36,10,36,0,0,5,31,5,21] >;

C2×C20.C23 in GAP, Magma, Sage, TeX

C_2\times C_{20}.C_2^3
% in TeX

G:=Group("C2xC20.C2^3");
// GroupNames label

G:=SmallGroup(320,1480);
// by ID

G=gap.SmallGroup(320,1480);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,136,1684,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^20=c^2=1,d^2=e^2=b^10,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,e*b*e^-1=b^11,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=b^10*d>;
// generators/relations

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