Copied to
clipboard

G = C10.2- 1+4order 320 = 26·5

3rd non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.2- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C10.2- 1+4
 Lower central C5 — C2×C10 — C10.2- 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C10.2- 1+4
G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=a5b2, e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a5b-1, dbd-1=a5b, be=eb, dcd-1=a5c, ce=ec, ede-1=b2d >

Subgroups: 1062 in 292 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×12], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×21], D4 [×14], Q8 [×4], C23, C23 [×3], D5 [×4], C10 [×3], C10 [×2], C42, C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4 [×6], C2×Q8 [×2], C4○D4 [×8], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×5], D10 [×2], D10 [×8], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C2×C4⋊C4, C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×10], D20 [×6], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×4], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×6], C22×D5, C22×D5 [×2], C22×C10, D46D4, C4×Dic5, C10.D4, C10.D4 [×4], C4⋊Dic5, D10⋊C4, D10⋊C4 [×6], C23.D5, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×4], C2×D20, C2×D20 [×2], C4○D20 [×8], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, C22×C20 [×2], C20⋊Q8, D5×C4⋊C4, D208C4, D10.13D4 [×2], C4⋊D20, D10⋊Q8 [×2], C4×C5⋊D4, C23.23D10 [×2], C207D4, C10×C4⋊C4, C2×C4○D20 [×2], C10.2- 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- 1+4, C22×D5 [×7], D46D4, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, Q8.10D10, C10.2- 1+4

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 C4○D20 2- 1+4 D4×D5 Q8.10D10 kernel C10.2- 1+4 C20⋊Q8 D5×C4⋊C4 D20⋊8C4 D10.13D4 C4⋊D20 D10⋊Q8 C4×C5⋊D4 C23.23D10 C20⋊7D4 C10×C4⋊C4 C2×C4○D20 C5⋊D4 C2×C4⋊C4 C20 C4⋊C4 C22×C4 C4 C10 C22 C2 # reps 1 1 1 1 2 1 2 1 2 1 1 2 4 2 4 8 6 16 1 4 4

Matrix representation of C10.2- 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 7 7 0 0 0 0 34 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 18 9 0 0 0 0 37 23 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 1 0
,
 32 0 0 0 0 0 36 9 0 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 23 32 0 0 0 0 27 18 0 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1
,
 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 0 1 0 0 0 0 40 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[18,37,0,0,0,0,9,23,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[32,36,0,0,0,0,0,9,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[23,27,0,0,0,0,32,18,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0] >;`

C10.2- 1+4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(320,1179);`
`// by ID`

`G=gap.SmallGroup(320,1179);`
`# by ID`

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

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

׿
×
𝔽