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## G = C10.732- 1+4order 320 = 26·5

### 28th 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 — C10.732- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C10.732- 1+4
 Lower central C5 — C2×C10 — C10.732- 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C10.732- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=a5b2d >

Subgroups: 982 in 292 conjugacy classes, 107 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, D46D4, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, D42D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, Dic5.14D4, D10.12D4, C20⋊Q8, D5×C4⋊C4, C2×C4⋊Dic5, C4×C5⋊D4, D4×Dic5, C23.18D10, C202D4, C5×C4⋊D4, C2×D42D5, C10.732- 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, D42D5, C23×D5, C2×D4×D5, C2×D42D5, D4.10D10, C10.732- 1+4

Smallest permutation representation of C10.732- 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 38 30 50)(2 37 21 49)(3 36 22 48)(4 35 23 47)(5 34 24 46)(6 33 25 45)(7 32 26 44)(8 31 27 43)(9 40 28 42)(10 39 29 41)(11 138 153 150)(12 137 154 149)(13 136 155 148)(14 135 156 147)(15 134 157 146)(16 133 158 145)(17 132 159 144)(18 131 160 143)(19 140 151 142)(20 139 152 141)(51 80 63 82)(52 79 64 81)(53 78 65 90)(54 77 66 89)(55 76 67 88)(56 75 68 87)(57 74 69 86)(58 73 70 85)(59 72 61 84)(60 71 62 83)(91 122 103 120)(92 121 104 119)(93 130 105 118)(94 129 106 117)(95 128 107 116)(96 127 108 115)(97 126 109 114)(98 125 110 113)(99 124 101 112)(100 123 102 111)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 41)(8 42)(9 43)(10 44)(11 133)(12 134)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 131)(20 132)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 31)(29 32)(30 33)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 81)(60 82)(61 79)(62 80)(63 71)(64 72)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(141 159)(142 160)(143 151)(144 152)(145 153)(146 154)(147 155)(148 156)(149 157)(150 158)
(1 53 30 65)(2 52 21 64)(3 51 22 63)(4 60 23 62)(5 59 24 61)(6 58 25 70)(7 57 26 69)(8 56 27 68)(9 55 28 67)(10 54 29 66)(11 130 153 118)(12 129 154 117)(13 128 155 116)(14 127 156 115)(15 126 157 114)(16 125 158 113)(17 124 159 112)(18 123 160 111)(19 122 151 120)(20 121 152 119)(31 75 43 87)(32 74 44 86)(33 73 45 85)(34 72 46 84)(35 71 47 83)(36 80 48 82)(37 79 49 81)(38 78 50 90)(39 77 41 89)(40 76 42 88)(91 140 103 142)(92 139 104 141)(93 138 105 150)(94 137 106 149)(95 136 107 148)(96 135 108 147)(97 134 109 146)(98 133 110 145)(99 132 101 144)(100 131 102 143)
(1 98 25 105)(2 99 26 106)(3 100 27 107)(4 91 28 108)(5 92 29 109)(6 93 30 110)(7 94 21 101)(8 95 22 102)(9 96 23 103)(10 97 24 104)(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), (1,38,30,50)(2,37,21,49)(3,36,22,48)(4,35,23,47)(5,34,24,46)(6,33,25,45)(7,32,26,44)(8,31,27,43)(9,40,28,42)(10,39,29,41)(11,138,153,150)(12,137,154,149)(13,136,155,148)(14,135,156,147)(15,134,157,146)(16,133,158,145)(17,132,159,144)(18,131,160,143)(19,140,151,142)(20,139,152,141)(51,80,63,82)(52,79,64,81)(53,78,65,90)(54,77,66,89)(55,76,67,88)(56,75,68,87)(57,74,69,86)(58,73,70,85)(59,72,61,84)(60,71,62,83)(91,122,103,120)(92,121,104,119)(93,130,105,118)(94,129,106,117)(95,128,107,116)(96,127,108,115)(97,126,109,114)(98,125,110,113)(99,124,101,112)(100,123,102,111), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,41)(8,42)(9,43)(10,44)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,131)(20,132)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,31)(29,32)(30,33)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,81)(60,82)(61,79)(62,80)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(141,159)(142,160)(143,151)(144,152)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158), (1,53,30,65)(2,52,21,64)(3,51,22,63)(4,60,23,62)(5,59,24,61)(6,58,25,70)(7,57,26,69)(8,56,27,68)(9,55,28,67)(10,54,29,66)(11,130,153,118)(12,129,154,117)(13,128,155,116)(14,127,156,115)(15,126,157,114)(16,125,158,113)(17,124,159,112)(18,123,160,111)(19,122,151,120)(20,121,152,119)(31,75,43,87)(32,74,44,86)(33,73,45,85)(34,72,46,84)(35,71,47,83)(36,80,48,82)(37,79,49,81)(38,78,50,90)(39,77,41,89)(40,76,42,88)(91,140,103,142)(92,139,104,141)(93,138,105,150)(94,137,106,149)(95,136,107,148)(96,135,108,147)(97,134,109,146)(98,133,110,145)(99,132,101,144)(100,131,102,143), (1,98,25,105)(2,99,26,106)(3,100,27,107)(4,91,28,108)(5,92,29,109)(6,93,30,110)(7,94,21,101)(8,95,22,102)(9,96,23,103)(10,97,24,104)(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), (1,38,30,50)(2,37,21,49)(3,36,22,48)(4,35,23,47)(5,34,24,46)(6,33,25,45)(7,32,26,44)(8,31,27,43)(9,40,28,42)(10,39,29,41)(11,138,153,150)(12,137,154,149)(13,136,155,148)(14,135,156,147)(15,134,157,146)(16,133,158,145)(17,132,159,144)(18,131,160,143)(19,140,151,142)(20,139,152,141)(51,80,63,82)(52,79,64,81)(53,78,65,90)(54,77,66,89)(55,76,67,88)(56,75,68,87)(57,74,69,86)(58,73,70,85)(59,72,61,84)(60,71,62,83)(91,122,103,120)(92,121,104,119)(93,130,105,118)(94,129,106,117)(95,128,107,116)(96,127,108,115)(97,126,109,114)(98,125,110,113)(99,124,101,112)(100,123,102,111), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,41)(8,42)(9,43)(10,44)(11,133)(12,134)(13,135)(14,136)(15,137)(16,138)(17,139)(18,140)(19,131)(20,132)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,31)(29,32)(30,33)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,81)(60,82)(61,79)(62,80)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(141,159)(142,160)(143,151)(144,152)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158), (1,53,30,65)(2,52,21,64)(3,51,22,63)(4,60,23,62)(5,59,24,61)(6,58,25,70)(7,57,26,69)(8,56,27,68)(9,55,28,67)(10,54,29,66)(11,130,153,118)(12,129,154,117)(13,128,155,116)(14,127,156,115)(15,126,157,114)(16,125,158,113)(17,124,159,112)(18,123,160,111)(19,122,151,120)(20,121,152,119)(31,75,43,87)(32,74,44,86)(33,73,45,85)(34,72,46,84)(35,71,47,83)(36,80,48,82)(37,79,49,81)(38,78,50,90)(39,77,41,89)(40,76,42,88)(91,140,103,142)(92,139,104,141)(93,138,105,150)(94,137,106,149)(95,136,107,148)(96,135,108,147)(97,134,109,146)(98,133,110,145)(99,132,101,144)(100,131,102,143), (1,98,25,105)(2,99,26,106)(3,100,27,107)(4,91,28,108)(5,92,29,109)(6,93,30,110)(7,94,21,101)(8,95,22,102)(9,96,23,103)(10,97,24,104)(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)], [(1,38,30,50),(2,37,21,49),(3,36,22,48),(4,35,23,47),(5,34,24,46),(6,33,25,45),(7,32,26,44),(8,31,27,43),(9,40,28,42),(10,39,29,41),(11,138,153,150),(12,137,154,149),(13,136,155,148),(14,135,156,147),(15,134,157,146),(16,133,158,145),(17,132,159,144),(18,131,160,143),(19,140,151,142),(20,139,152,141),(51,80,63,82),(52,79,64,81),(53,78,65,90),(54,77,66,89),(55,76,67,88),(56,75,68,87),(57,74,69,86),(58,73,70,85),(59,72,61,84),(60,71,62,83),(91,122,103,120),(92,121,104,119),(93,130,105,118),(94,129,106,117),(95,128,107,116),(96,127,108,115),(97,126,109,114),(98,125,110,113),(99,124,101,112),(100,123,102,111)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,41),(8,42),(9,43),(10,44),(11,133),(12,134),(13,135),(14,136),(15,137),(16,138),(17,139),(18,140),(19,131),(20,132),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39),(27,40),(28,31),(29,32),(30,33),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,81),(60,82),(61,79),(62,80),(63,71),(64,72),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,127),(108,128),(109,129),(110,130),(141,159),(142,160),(143,151),(144,152),(145,153),(146,154),(147,155),(148,156),(149,157),(150,158)], [(1,53,30,65),(2,52,21,64),(3,51,22,63),(4,60,23,62),(5,59,24,61),(6,58,25,70),(7,57,26,69),(8,56,27,68),(9,55,28,67),(10,54,29,66),(11,130,153,118),(12,129,154,117),(13,128,155,116),(14,127,156,115),(15,126,157,114),(16,125,158,113),(17,124,159,112),(18,123,160,111),(19,122,151,120),(20,121,152,119),(31,75,43,87),(32,74,44,86),(33,73,45,85),(34,72,46,84),(35,71,47,83),(36,80,48,82),(37,79,49,81),(38,78,50,90),(39,77,41,89),(40,76,42,88),(91,140,103,142),(92,139,104,141),(93,138,105,150),(94,137,106,149),(95,136,107,148),(96,135,108,147),(97,134,109,146),(98,133,110,145),(99,132,101,144),(100,131,102,143)], [(1,98,25,105),(2,99,26,106),(3,100,27,107),(4,91,28,108),(5,92,29,109),(6,93,30,110),(7,94,21,101),(8,95,22,102),(9,96,23,103),(10,97,24,104),(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)]])`

53 conjugacy classes

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

53 irreducible representations

 dim 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 D4 D5 C4○D4 D10 D10 D10 D10 2- 1+4 D4⋊2D5 D4×D5 D4.10D10 kernel C10.732- 1+4 Dic5.14D4 D10.12D4 C20⋊Q8 D5×C4⋊C4 C2×C4⋊Dic5 C4×C5⋊D4 D4×Dic5 C23.18D10 C20⋊2D4 C5×C4⋊D4 C2×D4⋊2D5 C5⋊D4 C4⋊D4 C20 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C4 C22 C2 # reps 1 2 2 1 1 1 1 1 2 1 1 2 4 2 4 4 2 2 6 1 4 4 4

Matrix representation of C10.732- 1+4 in GL6(𝔽41)

 6 7 0 0 0 0 35 0 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
,
 6 7 0 0 0 0 36 35 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 35 34 0 0 0 0 5 6 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 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 27 22 0 0 0 0 19 14

`G:=sub<GL(6,GF(41))| [6,35,0,0,0,0,7,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[6,36,0,0,0,0,7,35,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[35,5,0,0,0,0,34,6,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,27,19,0,0,0,0,22,14] >;`

C10.732- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{73}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.73ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1283);`
`// by ID`

`G=gap.SmallGroup(320,1283);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,675,185,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=b^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,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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