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## G = C10.822- 1+4order 320 = 26·5

### 37th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.822- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C2×C4○D20 — C10.822- 1+4
 Lower central C5 — C2×C10 — C10.822- 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C10.822- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=b2, bab-1=cac=dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=a5b2d >

Subgroups: 1046 in 292 conjugacy classes, 105 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4⋊Q8, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, D46D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, D42D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, Dic5.14D4, Dic54D4, D10.12D4, D10⋊D4, C22.D20, C20⋊Q8, D5×C4⋊C4, D10.13D4, D10⋊Q8, C2×C10.D4, Dic5⋊D4, C5×C22.D4, C2×C4○D20, C2×D42D5, C10.822- 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2- 1+4, C22×D5, D46D4, D4×D5, C23×D5, C2×D4×D5, D5×C4○D4, D4.10D10, C10.822- 1+4

Smallest permutation representation of C10.822- 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 58 23 65)(2 57 24 64)(3 56 25 63)(4 55 26 62)(5 54 27 61)(6 53 28 70)(7 52 29 69)(8 51 30 68)(9 60 21 67)(10 59 22 66)(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)(91 135 108 142)(92 134 109 141)(93 133 110 150)(94 132 101 149)(95 131 102 148)(96 140 103 147)(97 139 104 146)(98 138 105 145)(99 137 106 144)(100 136 107 143)
(1 138)(2 137)(3 136)(4 135)(5 134)(6 133)(7 132)(8 131)(9 140)(10 139)(11 33)(12 32)(13 31)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 147)(22 146)(23 145)(24 144)(25 143)(26 142)(27 141)(28 150)(29 149)(30 148)(41 157)(42 156)(43 155)(44 154)(45 153)(46 152)(47 151)(48 160)(49 159)(50 158)(51 95)(52 94)(53 93)(54 92)(55 91)(56 100)(57 99)(58 98)(59 97)(60 96)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 110)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)(81 124)(82 123)(83 122)(84 121)(85 130)(86 129)(87 128)(88 127)(89 126)(90 125)
(1 16 23 153)(2 15 24 152)(3 14 25 151)(4 13 26 160)(5 12 27 159)(6 11 28 158)(7 20 29 157)(8 19 30 156)(9 18 21 155)(10 17 22 154)(31 142 48 135)(32 141 49 134)(33 150 50 133)(34 149 41 132)(35 148 42 131)(36 147 43 140)(37 146 44 139)(38 145 45 138)(39 144 46 137)(40 143 47 136)(51 115 68 122)(52 114 69 121)(53 113 70 130)(54 112 61 129)(55 111 62 128)(56 120 63 127)(57 119 64 126)(58 118 65 125)(59 117 66 124)(60 116 67 123)(71 107 88 100)(72 106 89 99)(73 105 90 98)(74 104 81 97)(75 103 82 96)(76 102 83 95)(77 101 84 94)(78 110 85 93)(79 109 86 92)(80 108 87 91)
(1 58 23 65)(2 59 24 66)(3 60 25 67)(4 51 26 68)(5 52 27 69)(6 53 28 70)(7 54 29 61)(8 55 30 62)(9 56 21 63)(10 57 22 64)(11 125 158 118)(12 126 159 119)(13 127 160 120)(14 128 151 111)(15 129 152 112)(16 130 153 113)(17 121 154 114)(18 122 155 115)(19 123 156 116)(20 124 157 117)(31 88 48 71)(32 89 49 72)(33 90 50 73)(34 81 41 74)(35 82 42 75)(36 83 43 76)(37 84 44 77)(38 85 45 78)(39 86 46 79)(40 87 47 80)(91 148 108 131)(92 149 109 132)(93 150 110 133)(94 141 101 134)(95 142 102 135)(96 143 103 136)(97 144 104 137)(98 145 105 138)(99 146 106 139)(100 147 107 140)```

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 D10 2- 1+4 D4×D5 D5×C4○D4 D4.10D10 kernel C10.822- 1+4 Dic5.14D4 Dic5⋊4D4 D10.12D4 D10⋊D4 C22.D20 C20⋊Q8 D5×C4⋊C4 D10.13D4 D10⋊Q8 C2×C10.D4 Dic5⋊D4 C5×C22.D4 C2×C4○D20 C2×D4⋊2D5 C5⋊D4 C22.D4 Dic5 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C22 C2 C2 # reps 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 4 2 4 6 4 2 2 1 4 4 4

Matrix representation of C10.822- 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 35 0 0 0 0 6 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 6 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 9 0 0 0 0 32 0 0 0 0 0 0 0 1 0 0 0 0 0 6 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 35 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 32 0 0 0 0 0 0 32 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 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;`

C10.822- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{82}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.82ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1327);`
`// by ID`

`G=gap.SmallGroup(320,1327);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,184,570,185,136,12550]);`
`// Polycyclic`

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

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