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## G = C10.662+ 1+4order 320 = 26·5

### 66th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.662+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.13D4 — C10.662+ 1+4
 Lower central C5 — C2×C10 — C10.662+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C10.662+ 1+4
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, ebe=a5b, cd=dc, ce=ec, ede=b2d >

Subgroups: 974 in 240 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C4 [×11], C22, C22 [×15], C5, C2×C4 [×5], C2×C4 [×11], D4 [×12], C23 [×2], C23 [×3], D5 [×3], C10 [×3], C10 [×2], C42 [×2], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×9], Dic5 [×2], Dic5 [×4], C20 [×5], D10 [×9], C2×C10, C2×C10 [×6], C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4, C22.D4 [×3], C42.C2, C41D4, C4×D5 [×4], D20 [×3], C2×Dic5 [×5], C2×Dic5, C5⋊D4 [×8], C2×C20 [×5], C2×C20, C5×D4, C22×D5 [×3], C22×C10 [×2], C22.34C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5, C5×C22⋊C4 [×3], C5×C4⋊C4 [×2], C2×C4×D5 [×3], C2×D20 [×3], C22×Dic5, C2×C5⋊D4 [×6], C22×C20, D4×C10, Dic54D4, D10.12D4, D10⋊D4 [×3], C22.D20, Dic5.Q8, C4⋊C47D5, D10.13D4, C4⋊D20, C4×C5⋊D4, C207D4, Dic5⋊D4, C20⋊D4, C5×C22.D4, C10.662+ 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4 [×2], C22×D5 [×7], C22.34C24, C23×D5, D46D10, D5×C4○D4, D48D10, C10.662+ 1+4

Smallest permutation representation of C10.662+ 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)
(1 133 13 123)(2 134 14 124)(3 135 15 125)(4 136 16 126)(5 137 17 127)(6 138 18 128)(7 139 19 129)(8 140 20 130)(9 131 11 121)(10 132 12 122)(21 151 31 141)(22 152 32 142)(23 153 33 143)(24 154 34 144)(25 155 35 145)(26 156 36 146)(27 157 37 147)(28 158 38 148)(29 159 39 149)(30 160 40 150)(41 96 51 86)(42 97 52 87)(43 98 53 88)(44 99 54 89)(45 100 55 90)(46 91 56 81)(47 92 57 82)(48 93 58 83)(49 94 59 84)(50 95 60 85)(61 116 71 106)(62 117 72 107)(63 118 73 108)(64 119 74 109)(65 120 75 110)(66 111 76 101)(67 112 77 102)(68 113 78 103)(69 114 79 104)(70 115 80 105)
(1 88 6 83)(2 89 7 84)(3 90 8 85)(4 81 9 86)(5 82 10 87)(11 96 16 91)(12 97 17 92)(13 98 18 93)(14 99 19 94)(15 100 20 95)(21 106 26 101)(22 107 27 102)(23 108 28 103)(24 109 29 104)(25 110 30 105)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)
(1 138 13 128)(2 137 14 127)(3 136 15 126)(4 135 16 125)(5 134 17 124)(6 133 18 123)(7 132 19 122)(8 131 20 121)(9 140 11 130)(10 139 12 129)(21 145 31 155)(22 144 32 154)(23 143 33 153)(24 142 34 152)(25 141 35 151)(26 150 36 160)(27 149 37 159)(28 148 38 158)(29 147 39 157)(30 146 40 156)(41 85 51 95)(42 84 52 94)(43 83 53 93)(44 82 54 92)(45 81 55 91)(46 90 56 100)(47 89 57 99)(48 88 58 98)(49 87 59 97)(50 86 60 96)(61 120 71 110)(62 119 72 109)(63 118 73 108)(64 117 74 107)(65 116 75 106)(66 115 76 105)(67 114 77 104)(68 113 78 103)(69 112 79 102)(70 111 80 101)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 146)(122 147)(123 148)(124 149)(125 150)(126 141)(127 142)(128 143)(129 144)(130 145)(131 156)(132 157)(133 158)(134 159)(135 160)(136 151)(137 152)(138 153)(139 154)(140 155)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4⋊6D10 D5×C4○D4 D4⋊8D10 kernel C10.662+ 1+4 Dic5⋊4D4 D10.12D4 D10⋊D4 C22.D20 Dic5.Q8 C4⋊C4⋊7D5 D10.13D4 C4⋊D20 C4×C5⋊D4 C20⋊7D4 Dic5⋊D4 C20⋊D4 C5×C22.D4 C22.D4 Dic5 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C2 C2 C2 # reps 1 1 1 3 1 1 1 1 1 1 1 1 1 1 2 4 6 4 2 2 2 4 4 4

Matrix representation of C10.662+ 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 6 0 0 0 0 34 7 0 0 0 0 6 0 6 6 0 0 1 35 35 1
,
 1 0 0 0 0 0 21 40 0 0 0 0 0 0 21 0 14 27 0 0 39 35 28 39 0 0 18 38 6 0 0 0 32 38 27 20
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 4 13 15 15 0 0 16 11 0 38 0 0 28 0 39 28 0 0 39 28 39 28
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 26 6 29 27 0 0 38 1 25 39 0 0 23 3 35 0 0 0 1 9 17 20
,
 1 37 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 1 0 0 6 40 39 6 0 0 0 0 1 0 0 0 1 0 1 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,34,6,1,0,0,6,7,0,35,0,0,0,0,6,35,0,0,0,0,6,1],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,21,39,18,32,0,0,0,35,38,38,0,0,14,28,6,27,0,0,27,39,0,20],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,4,16,28,39,0,0,13,11,0,28,0,0,15,0,39,39,0,0,15,38,28,28],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,26,38,23,1,0,0,6,1,3,9,0,0,29,25,35,17,0,0,27,39,0,20],[1,0,0,0,0,0,37,40,0,0,0,0,0,0,0,6,0,1,0,0,0,40,0,0,0,0,40,39,1,1,0,0,1,6,0,0] >;`

C10.662+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{66}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.66ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1335);`
`// by ID`

`G=gap.SmallGroup(320,1335);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,184,675,297,12550]);`
`// Polycyclic`

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

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