Copied to
clipboard

## G = C10.682- 1+4order 320 = 26·5

### 23rd 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 — C10.682- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×D4⋊2D5 — C10.682- 1+4
 Lower central C5 — C2×C10 — C10.682- 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C10.682- 1+4
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, ab=ba, cac-1=eae=a-1, ad=da, cbc-1=a5b-1, dbd-1=a5b, be=eb, dcd-1=ece=a5c, ede=a5b2d >

Subgroups: 1102 in 294 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C4 [×12], C22, C22 [×2], C22 [×14], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×20], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×7], C22×C4, C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], Dic5 [×4], Dic5 [×4], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C4⋊C4, C4⋊D4, C4⋊D4 [×7], C22⋊Q8 [×4], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], D20, C2×Dic5 [×10], C2×Dic5 [×5], C5⋊D4 [×10], C2×C20 [×2], C2×C20 [×2], C2×C20, C5×D4 [×5], C22×D5 [×2], C22×C10, C22×C10 [×2], C22.31C24, C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, D42D5 [×8], C22×Dic5 [×2], C22×Dic5 [×2], C2×C5⋊D4 [×6], C22×C20, D4×C10, D4×C10 [×2], Dic5.14D4 [×2], D10⋊D4 [×2], D10⋊Q8 [×2], C2×C10.D4, C207D4, Dic5⋊D4 [×4], C5×C4⋊D4, C2×D42D5 [×2], C10.682- 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.31C24, D4×D5 [×2], C23×D5, C2×D4×D5, D46D10, D4.10D10, C10.682- 1+4

Smallest permutation representation of C10.682- 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)
(1 138 30 150)(2 139 21 141)(3 140 22 142)(4 131 23 143)(5 132 24 144)(6 133 25 145)(7 134 26 146)(8 135 27 147)(9 136 28 148)(10 137 29 149)(11 33 153 45)(12 34 154 46)(13 35 155 47)(14 36 156 48)(15 37 157 49)(16 38 158 50)(17 39 159 41)(18 40 160 42)(19 31 151 43)(20 32 152 44)(51 91 63 103)(52 92 64 104)(53 93 65 105)(54 94 66 106)(55 95 67 107)(56 96 68 108)(57 97 69 109)(58 98 70 110)(59 99 61 101)(60 100 62 102)(71 111 83 123)(72 112 84 124)(73 113 85 125)(74 114 86 126)(75 115 87 127)(76 116 88 128)(77 117 89 129)(78 118 90 130)(79 119 81 121)(80 120 82 122)
(1 45 6 50)(2 44 7 49)(3 43 8 48)(4 42 9 47)(5 41 10 46)(11 150 16 145)(12 149 17 144)(13 148 18 143)(14 147 19 142)(15 146 20 141)(21 32 26 37)(22 31 27 36)(23 40 28 35)(24 39 29 34)(25 38 30 33)(51 75 56 80)(52 74 57 79)(53 73 58 78)(54 72 59 77)(55 71 60 76)(61 89 66 84)(62 88 67 83)(63 87 68 82)(64 86 69 81)(65 85 70 90)(91 122 96 127)(92 121 97 126)(93 130 98 125)(94 129 99 124)(95 128 100 123)(101 112 106 117)(102 111 107 116)(103 120 108 115)(104 119 109 114)(105 118 110 113)(131 155 136 160)(132 154 137 159)(133 153 138 158)(134 152 139 157)(135 151 140 156)
(1 125 30 113)(2 126 21 114)(3 127 22 115)(4 128 23 116)(5 129 24 117)(6 130 25 118)(7 121 26 119)(8 122 27 120)(9 123 28 111)(10 124 29 112)(11 65 153 53)(12 66 154 54)(13 67 155 55)(14 68 156 56)(15 69 157 57)(16 70 158 58)(17 61 159 59)(18 62 160 60)(19 63 151 51)(20 64 152 52)(31 108 43 96)(32 109 44 97)(33 110 45 98)(34 101 46 99)(35 102 47 100)(36 103 48 91)(37 104 49 92)(38 105 50 93)(39 106 41 94)(40 107 42 95)(71 143 83 131)(72 144 84 132)(73 145 85 133)(74 146 86 134)(75 147 87 135)(76 148 88 136)(77 149 89 137)(78 150 90 138)(79 141 81 139)(80 142 82 140)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 158)(12 157)(13 156)(14 155)(15 154)(16 153)(17 152)(18 151)(19 160)(20 159)(31 42)(32 41)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(51 55)(52 54)(56 60)(57 59)(61 69)(62 68)(63 67)(64 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 84)(82 83)(85 90)(86 89)(87 88)(91 95)(92 94)(96 100)(97 99)(101 109)(102 108)(103 107)(104 106)(111 120)(112 119)(113 118)(114 117)(115 116)(121 124)(122 123)(125 130)(126 129)(127 128)(131 147)(132 146)(133 145)(134 144)(135 143)(136 142)(137 141)(138 150)(139 149)(140 148)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 D10 D10 D10 D10 2+ 1+4 2- 1+4 D4×D5 D4⋊6D10 D4.10D10 kernel C10.682- 1+4 Dic5.14D4 D10⋊D4 D10⋊Q8 C2×C10.D4 C20⋊7D4 Dic5⋊D4 C5×C4⋊D4 C2×D4⋊2D5 C2×Dic5 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C10 C22 C2 C2 # reps 1 2 2 2 1 1 4 1 2 4 2 4 2 2 6 1 1 4 4 4

Matrix representation of C10.682- 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 6 6 0 0 0 0 35 1 0 0 0 0 0 0 6 6 0 0 0 0 35 1
,
 1 0 0 0 0 0 1 40 0 0 0 0 0 0 0 0 39 28 0 0 0 0 13 2 0 0 39 28 0 0 0 0 13 2 0 0
,
 40 2 0 0 0 0 40 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 35 1 0 0 1 0 0 0 0 0 6 40 0 0
,
 40 2 0 0 0 0 0 1 0 0 0 0 0 0 34 0 28 18 0 0 0 34 23 13 0 0 13 23 7 0 0 0 18 28 0 7
,
 1 0 0 0 0 0 1 40 0 0 0 0 0 0 40 0 0 0 0 0 35 1 0 0 0 0 0 0 1 0 0 0 0 0 6 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,0,0,39,13,0,0,0,0,28,2,0,0,39,13,0,0,0,0,28,2,0,0],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,40,35,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,34,0,13,18,0,0,0,34,23,28,0,0,28,23,7,0,0,0,18,13,0,7],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;`

C10.682- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{68}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.68ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1269);`
`// by ID`

`G=gap.SmallGroup(320,1269);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,1123,570,185,12550]);`
`// 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,c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=a^5*b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=e*c*e=a^5*c,e*d*e=a^5*b^2*d>;`
`// generators/relations`

׿
×
𝔽