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G = C10.522+ 1+4order 320 = 26·5

52nd non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.522+ 1+4, C5⋊D43Q8, C4⋊C4.97D10, C56(D43Q8), C22⋊Q813D5, C22.2(Q8×D5), D10.22(C2×Q8), (C2×Q8).75D10, D103Q818C2, D10⋊Q822C2, (C2×C20).59C23, C22⋊C4.61D10, Dic5.24(C2×Q8), Dic5⋊Q816C2, Dic53Q827C2, C10.38(C22×Q8), (C2×C10).180C24, Dic54D4.3C2, (C22×C4).242D10, C2.54(D46D10), Dic5.40(C4○D4), Dic5.Q819C2, C4⋊Dic5.217C22, (Q8×C10).111C22, (C2×Dic5).91C23, C22.201(C23×D5), C23.193(C22×D5), Dic5.14D426C2, D10⋊C4.25C22, (C22×C10).208C23, (C22×C20).380C22, (C4×Dic5).117C22, C10.D4.30C22, (C22×D5).212C23, C23.D5.120C22, (C2×Dic10).167C22, (C22×Dic5).121C22, (D5×C4⋊C4)⋊28C2, C2.21(C2×Q8×D5), C2.51(D5×C4○D4), (C2×C10).9(C2×Q8), (C4×C5⋊D4).18C2, (C5×C22⋊Q8)⋊16C2, C10.163(C2×C4○D4), (C2×C4×D5).109C22, (C2×C4).50(C22×D5), (C2×C10.D4)⋊41C2, (C5×C4⋊C4).162C22, (C2×C5⋊D4).135C22, (C5×C22⋊C4).35C22, SmallGroup(320,1308)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.522+ 1+4
C1C5C10C2×C10C22×D5C2×C5⋊D4C4×C5⋊D4 — C10.522+ 1+4
C5C2×C10 — C10.522+ 1+4
C1C22C22⋊Q8

Generators and relations for C10.522+ 1+4
 G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, ab=ba, ac=ca, dad-1=eae=a-1, cbc-1=b-1, dbd-1=a5b, be=eb, cd=dc, ce=ec, ede=a5b2d >

Subgroups: 742 in 228 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×15], C22, C22 [×2], C22 [×6], C5, C2×C4 [×6], C2×C4 [×15], D4 [×4], Q8 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4, C22×C4 [×5], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×4], Dic5 [×5], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8, C22⋊Q8 [×5], C42.C2 [×2], C4⋊Q8, Dic10 [×3], C4×D5 [×4], C2×Dic5 [×7], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×6], C2×C20, C5×Q8, C22×D5, C22×C10, D43Q8, C4×Dic5 [×3], C10.D4 [×11], C4⋊Dic5 [×2], D10⋊C4 [×3], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4 [×3], C2×Dic10 [×2], C2×C4×D5 [×3], C22×Dic5 [×2], C2×C5⋊D4, C22×C20, Q8×C10, Dic5.14D4 [×2], Dic54D4 [×2], Dic53Q8, Dic5.Q8 [×2], D5×C4⋊C4, D10⋊Q8 [×2], C2×C10.D4, C4×C5⋊D4, Dic5⋊Q8, D103Q8, C5×C22⋊Q8, C10.522+ 1+4
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2+ 1+4, C22×D5 [×7], D43Q8, Q8×D5 [×2], C23×D5, D46D10, C2×Q8×D5, D5×C4○D4, C10.522+ 1+4

Smallest permutation representation of C10.522+ 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 90 25 75)(2 81 26 76)(3 82 27 77)(4 83 28 78)(5 84 29 79)(6 85 30 80)(7 86 21 71)(8 87 22 72)(9 88 23 73)(10 89 24 74)(11 100 160 105)(12 91 151 106)(13 92 152 107)(14 93 153 108)(15 94 154 109)(16 95 155 110)(17 96 156 101)(18 97 157 102)(19 98 158 103)(20 99 159 104)(31 66 46 51)(32 67 47 52)(33 68 48 53)(34 69 49 54)(35 70 50 55)(36 61 41 56)(37 62 42 57)(38 63 43 58)(39 64 44 59)(40 65 45 60)(111 136 126 141)(112 137 127 142)(113 138 128 143)(114 139 129 144)(115 140 130 145)(116 131 121 146)(117 132 122 147)(118 133 123 148)(119 134 124 149)(120 135 125 150)
(1 55 6 60)(2 56 7 51)(3 57 8 52)(4 58 9 53)(5 59 10 54)(11 125 16 130)(12 126 17 121)(13 127 18 122)(14 128 19 123)(15 129 20 124)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)(41 86 46 81)(42 87 47 82)(43 88 48 83)(44 89 49 84)(45 90 50 85)(91 136 96 131)(92 137 97 132)(93 138 98 133)(94 139 99 134)(95 140 100 135)(101 146 106 141)(102 147 107 142)(103 148 108 143)(104 149 109 144)(105 150 110 145)(111 156 116 151)(112 157 117 152)(113 158 118 153)(114 159 119 154)(115 160 120 155)
(1 135 25 150)(2 134 26 149)(3 133 27 148)(4 132 28 147)(5 131 29 146)(6 140 30 145)(7 139 21 144)(8 138 22 143)(9 137 23 142)(10 136 24 141)(11 35 160 50)(12 34 151 49)(13 33 152 48)(14 32 153 47)(15 31 154 46)(16 40 155 45)(17 39 156 44)(18 38 157 43)(19 37 158 42)(20 36 159 41)(51 99 66 104)(52 98 67 103)(53 97 68 102)(54 96 69 101)(55 95 70 110)(56 94 61 109)(57 93 62 108)(58 92 63 107)(59 91 64 106)(60 100 65 105)(71 119 86 124)(72 118 87 123)(73 117 88 122)(74 116 89 121)(75 115 90 130)(76 114 81 129)(77 113 82 128)(78 112 83 127)(79 111 84 126)(80 120 85 125)
(2 10)(3 9)(4 8)(5 7)(11 155)(12 154)(13 153)(14 152)(15 151)(16 160)(17 159)(18 158)(19 157)(20 156)(21 29)(22 28)(23 27)(24 26)(31 39)(32 38)(33 37)(34 36)(41 49)(42 48)(43 47)(44 46)(51 59)(52 58)(53 57)(54 56)(61 69)(62 68)(63 67)(64 66)(71 79)(72 78)(73 77)(74 76)(81 89)(82 88)(83 87)(84 86)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(100 110)(111 129)(112 128)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(120 130)(131 149)(132 148)(133 147)(134 146)(135 145)(136 144)(137 143)(138 142)(139 141)(140 150)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C···4G4H···4M4N4O4P4Q5A5B10A···10F10G10H10I10J20A···20H20I···20P
order12222222444···44···444445510···101010101020···2020···20
size1111221010224···410···1020202020222···244444···48···8

53 irreducible representations

dim11111111111122222224444
type++++++++++++-++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2Q8D5C4○D4D10D10D10D102+ 1+4Q8×D5D46D10D5×C4○D4
kernelC10.522+ 1+4Dic5.14D4Dic54D4Dic53Q8Dic5.Q8D5×C4⋊C4D10⋊Q8C2×C10.D4C4×C5⋊D4Dic5⋊Q8D103Q8C5×C22⋊Q8C5⋊D4C22⋊Q8Dic5C22⋊C4C4⋊C4C22×C4C2×Q8C10C22C2C2
# reps12212121111142446221444

Matrix representation of C10.522+ 1+4 in GL6(𝔽41)

4000000
0400000
007700
00344000
0000400
0000040
,
4000000
3510000
001000
000100
0000346
0000197
,
3200000
0320000
0040000
0004000
00002138
00003820
,
6390000
38350000
0040000
007100
0000203
0000321
,
100000
6400000
001000
00344000
000010
000001

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],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,19,0,0,0,0,6,7],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,21,38,0,0,0,0,38,20],[6,38,0,0,0,0,39,35,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,20,3,0,0,0,0,3,21],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C10.522+ 1+4 in GAP, Magma, Sage, TeX

C_{10}._{52}2_+^{1+4}
% in TeX

G:=Group("C10.52ES+(2,2)");
// GroupNames label

G:=SmallGroup(320,1308);
// by ID

G=gap.SmallGroup(320,1308);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,570,409,80,12550]);
// 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=e*a*e=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^5*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=a^5*b^2*d>;
// generators/relations

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