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## G = C10.362+ 1+4order 320 = 26·5

### 36th 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 — C10.362+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — D4×Dic5 — C10.362+ 1+4
 Lower central C5 — C2×C10 — C10.362+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C10.362+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=a5b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=a5b-1, dbd-1=a5b, be=eb, dcd-1=a5c, ce=ec, ede-1=a5b2d >

Subgroups: 670 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C422C2, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22.36C24, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, D4×C10, D4×C10, Dic5.14D4, C23.D10, Dic53Q8, C20⋊Q8, C20.48D4, C23.21D10, D4×Dic5, C23.18D10, C20.17D4, C20.17D4, C5×C4⋊D4, C10.362+ 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.36C24, D42D5, C23×D5, C2×D42D5, D46D10, D4.10D10, C10.362+ 1+4

Smallest permutation representation of C10.362+ 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)
(1 157 25 20)(2 156 26 19)(3 155 27 18)(4 154 28 17)(5 153 29 16)(6 152 30 15)(7 151 21 14)(8 160 22 13)(9 159 23 12)(10 158 24 11)(31 134 48 142)(32 133 49 141)(33 132 50 150)(34 131 41 149)(35 140 42 148)(36 139 43 147)(37 138 44 146)(38 137 45 145)(39 136 46 144)(40 135 47 143)(51 127 68 119)(52 126 69 118)(53 125 70 117)(54 124 61 116)(55 123 62 115)(56 122 63 114)(57 121 64 113)(58 130 65 112)(59 129 66 111)(60 128 67 120)(71 99 89 107)(72 98 90 106)(73 97 81 105)(74 96 82 104)(75 95 83 103)(76 94 84 102)(77 93 85 101)(78 92 86 110)(79 91 87 109)(80 100 88 108)
(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 159)(18 160)(19 151)(20 152)(31 48)(32 49)(33 50)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(51 68)(52 69)(53 70)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 101)(99 102)(100 103)(111 116)(112 117)(113 118)(114 119)(115 120)(121 126)(122 127)(123 128)(124 129)(125 130)(131 136)(132 137)(133 138)(134 139)(135 140)(141 146)(142 147)(143 148)(144 149)(145 150)
(1 137 30 150)(2 136 21 149)(3 135 22 148)(4 134 23 147)(5 133 24 146)(6 132 25 145)(7 131 26 144)(8 140 27 143)(9 139 28 142)(10 138 29 141)(11 32 153 44)(12 31 154 43)(13 40 155 42)(14 39 156 41)(15 38 157 50)(16 37 158 49)(17 36 159 48)(18 35 160 47)(19 34 151 46)(20 33 152 45)(51 94 63 107)(52 93 64 106)(53 92 65 105)(54 91 66 104)(55 100 67 103)(56 99 68 102)(57 98 69 101)(58 97 70 110)(59 96 61 109)(60 95 62 108)(71 122 84 119)(72 121 85 118)(73 130 86 117)(74 129 87 116)(75 128 88 115)(76 127 89 114)(77 126 90 113)(78 125 81 112)(79 124 82 111)(80 123 83 120)
(1 86 30 73)(2 87 21 74)(3 88 22 75)(4 89 23 76)(5 90 24 77)(6 81 25 78)(7 82 26 79)(8 83 27 80)(9 84 28 71)(10 85 29 72)(11 93 153 106)(12 94 154 107)(13 95 155 108)(14 96 156 109)(15 97 157 110)(16 98 158 101)(17 99 159 102)(18 100 160 103)(19 91 151 104)(20 92 152 105)(31 51 43 63)(32 52 44 64)(33 53 45 65)(34 54 46 66)(35 55 47 67)(36 56 48 68)(37 57 49 69)(38 58 50 70)(39 59 41 61)(40 60 42 62)(111 131 124 144)(112 132 125 145)(113 133 126 146)(114 134 127 147)(115 135 128 148)(116 136 129 149)(117 137 130 150)(118 138 121 141)(119 139 122 142)(120 140 123 143)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J ··· 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 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 4 4 4 2 2 4 4 4 10 10 10 10 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 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 2- 1+4 D4⋊2D5 D4⋊6D10 D4.10D10 kernel C10.362+ 1+4 Dic5.14D4 C23.D10 Dic5⋊3Q8 C20⋊Q8 C20.48D4 C23.21D10 D4×Dic5 C23.18D10 C20.17D4 C5×C4⋊D4 C4⋊D4 C20 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C10 C4 C2 C2 # reps 1 2 2 1 1 1 1 1 2 3 1 2 4 4 2 2 6 1 1 4 4 4

Matrix representation of C10.362+ 1+4 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 35 35 0 0 0 0 0 0 6 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 32 0 0 0 0 0 0 0 7 9 0 0 0 0 0 0 0 0 3 5 0 0 0 0 0 0 23 38 0 0 0 0 0 0 0 0 34 1 1 13 0 0 0 0 28 26 13 13 0 0 0 0 3 25 11 17 0 0 0 0 16 25 5 11
,
 1 0 0 0 0 0 0 0 22 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 30 40 0 0 0 0 0 1 7 0 40
,
 9 29 0 0 0 0 0 0 34 32 0 0 0 0 0 0 0 0 3 5 0 0 0 0 0 0 23 38 0 0 0 0 0 0 0 0 34 27 13 1 0 0 0 0 28 26 13 13 0 0 0 0 3 19 17 11 0 0 0 0 16 19 11 5
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 34 7 0 0 0 0 0 0 28 7 0 0 0 0 0 0 10 26 20 14 0 0 0 0 9 21 27 21

`G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,35,6,0,0,0,0,0,0,35,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[32,7,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,23,0,0,0,0,0,0,5,38,0,0,0,0,0,0,0,0,34,28,3,16,0,0,0,0,1,26,25,25,0,0,0,0,1,13,11,5,0,0,0,0,13,13,17,11],[1,22,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,30,7,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[9,34,0,0,0,0,0,0,29,32,0,0,0,0,0,0,0,0,3,23,0,0,0,0,0,0,5,38,0,0,0,0,0,0,0,0,34,28,3,16,0,0,0,0,27,26,19,19,0,0,0,0,13,13,17,11,0,0,0,0,1,13,11,5],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,34,28,10,9,0,0,0,0,7,7,26,21,0,0,0,0,0,0,20,27,0,0,0,0,0,0,14,21] >;`

C10.362+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{36}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.36ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1275);`
`// by ID`

`G=gap.SmallGroup(320,1275);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,219,675,570,297,12550]);`
`// Polycyclic`

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

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