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## G = (C5×D4).D4order 320 = 26·5

### 8th non-split extension by C5×D4 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C5×D4).D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — D4×Dic5 — (C5×D4).D4
 Lower central C5 — C10 — C2×C20 — (C5×D4).D4
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for (C5×D4).D4
G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, cbc=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=b2d-1 >

Subgroups: 534 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C52C8, C40, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×C10, D4.2D4, C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4, D4⋊D5, C23.D5, C2×C40, C5×SD16, C2×D20, C22×Dic5, D4×C10, Q8×C10, C20.8Q8, D205C4, Q8⋊Dic5, C2×D4⋊D5, D4×Dic5, C20.23D4, C10×SD16, (C5×D4).D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, C5⋊D4, C22×D5, D4.2D4, D4×D5, D42D5, C2×C5⋊D4, D40⋊C2, SD163D5, Dic5⋊D4, (C5×D4).D4

Smallest permutation representation of (C5×D4).D4
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 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 80)(81 101 86 106)(82 102 87 107)(83 103 88 108)(84 104 89 109)(85 105 90 110)(91 111 96 116)(92 112 97 117)(93 113 98 118)(94 114 99 119)(95 115 100 120)(121 146 126 141)(122 147 127 142)(123 148 128 143)(124 149 129 144)(125 150 130 145)(131 156 136 151)(132 157 137 152)(133 158 138 153)(134 159 139 154)(135 160 140 155)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 101)(7 102)(8 103)(9 104)(10 105)(11 116)(12 117)(13 118)(14 119)(15 120)(16 111)(17 112)(18 113)(19 114)(20 115)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 141)(42 142)(43 143)(44 144)(45 145)(46 146)(47 147)(48 148)(49 149)(50 150)(51 151)(52 152)(53 153)(54 154)(55 155)(56 156)(57 157)(58 158)(59 159)(60 160)(61 126)(62 127)(63 128)(64 129)(65 130)(66 121)(67 122)(68 123)(69 124)(70 125)(71 136)(72 137)(73 138)(74 139)(75 140)(76 131)(77 132)(78 133)(79 134)(80 135)
(1 41 11 51)(2 45 12 55)(3 44 13 54)(4 43 14 53)(5 42 15 52)(6 46 16 56)(7 50 17 60)(8 49 18 59)(9 48 19 58)(10 47 20 57)(21 66 31 76)(22 70 32 80)(23 69 33 79)(24 68 34 78)(25 67 35 77)(26 61 36 71)(27 65 37 75)(28 64 38 74)(29 63 39 73)(30 62 40 72)(81 121 91 131)(82 125 92 135)(83 124 93 134)(84 123 94 133)(85 122 95 132)(86 126 96 136)(87 130 97 140)(88 129 98 139)(89 128 99 138)(90 127 100 137)(101 146 111 156)(102 150 112 160)(103 149 113 159)(104 148 114 158)(105 147 115 157)(106 141 116 151)(107 145 117 155)(108 144 118 154)(109 143 119 153)(110 142 120 152)
(2 5)(3 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 56)(42 60)(43 59)(44 58)(45 57)(46 51)(47 55)(48 54)(49 53)(50 52)(61 71)(62 75)(63 74)(64 73)(65 72)(66 76)(67 80)(68 79)(69 78)(70 77)(81 106)(82 110)(83 109)(84 108)(85 107)(86 101)(87 105)(88 104)(89 103)(90 102)(91 116)(92 120)(93 119)(94 118)(95 117)(96 111)(97 115)(98 114)(99 113)(100 112)(121 156)(122 160)(123 159)(124 158)(125 157)(126 151)(127 155)(128 154)(129 153)(130 152)(131 146)(132 150)(133 149)(134 148)(135 147)(136 141)(137 145)(138 144)(139 143)(140 142)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 40 2 2 8 10 10 20 20 20 2 2 4 4 20 20 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 C4○D8 C5⋊D4 C8⋊C22 D4⋊2D5 D4×D5 D40⋊C2 SD16⋊3D5 kernel (C5×D4).D4 C20.8Q8 D20⋊5C4 Q8⋊Dic5 C2×D4⋊D5 D4×Dic5 C20.23D4 C10×SD16 C2×Dic5 C5×D4 C2×SD16 C20 C2×C8 C2×D4 C2×Q8 C10 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 1 2 2 4 4

Matrix representation of (C5×D4).D4 in GL6(𝔽41)

 0 40 0 0 0 0 1 6 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
,
 1 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 5 0 0 0 0 16 40
,
 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 24 19 0 0 0 0 28 17
,
 6 35 0 0 0 0 40 35 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 6 35 0 0 0 0 40 35 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 16 40

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,5,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,28,0,0,0,0,19,17],[6,40,0,0,0,0,35,35,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[6,40,0,0,0,0,35,35,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,16,0,0,0,0,0,40] >;

(C5×D4).D4 in GAP, Magma, Sage, TeX

(C_5\times D_4).D_4
% in TeX

G:=Group("(C5xD4).D4");
// GroupNames label

G:=SmallGroup(320,792);
// by ID

G=gap.SmallGroup(320,792);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1094,135,184,570,297,136,12550]);
// Polycyclic

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

׿
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