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G = C10.1152+ 1+4order 320 = 26·5

24th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.1152+ 1+4, C4⋊D421D5, C202D426C2, C4⋊C4.184D10, (D4×Dic5)⋊25C2, (C2×D4).157D10, (C2×C20).44C23, C22⋊C4.10D10, C4.Dic1020C2, Dic5⋊D416C2, C20.205(C4○D4), C4.98(D42D5), (C2×C10).162C24, (C22×C4).229D10, C2.30(D48D10), C23.22(C22×D5), (D4×C10).127C22, C23.D1020C2, C22.D2012C2, C22.2(D42D5), C4⋊Dic5.209C22, (C22×D5).69C23, C22.183(C23×D5), C23.D5.28C22, (C22×C20).245C22, (C22×C10).190C23, C58(C22.47C24), (C4×Dic5).106C22, (C2×Dic5).240C23, C10.D4.20C22, D10⋊C4.106C22, (C22×Dic5).114C22, (C4×C5⋊D4)⋊21C2, C4⋊C47D522C2, (C5×C4⋊D4)⋊24C2, (C2×C4⋊Dic5)⋊41C2, C10.86(C2×C4○D4), (C2×C4×D5).97C22, C2.41(C2×D42D5), (C2×C10).24(C4○D4), (C5×C4⋊C4).149C22, (C2×C4).589(C22×D5), (C2×C5⋊D4).127C22, (C5×C22⋊C4).18C22, SmallGroup(320,1290)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.1152+ 1+4
C1C5C10C2×C10C22×D5C2×C5⋊D4C4×C5⋊D4 — C10.1152+ 1+4
C5C2×C10 — C10.1152+ 1+4
C1C22C4⋊D4

Generators and relations for C10.1152+ 1+4
 G = < a,b,c,d,e | a10=b4=1, c2=e2=a5, d2=b2, ab=ba, cac-1=dad-1=eae-1=a-1, cbc-1=b-1, dbd-1=a5b, be=eb, cd=dc, ce=ec, ede-1=a5b2d >

Subgroups: 790 in 238 conjugacy classes, 99 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×10], C23, C23 [×2], C23, D5, C10 [×3], C10 [×4], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×3], Dic5 [×7], C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4, C4⋊D4 [×3], C22.D4 [×2], C42.C2, C422C2 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×D5, C22×C10, C22×C10 [×2], C22.47C24, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4, D10⋊C4 [×2], C23.D5, C23.D5 [×4], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C22×Dic5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, D4×C10, D4×C10 [×2], C23.D10 [×2], C22.D20 [×2], C4.Dic10, C4⋊C47D5, C2×C4⋊Dic5, C4×C5⋊D4, D4×Dic5, D4×Dic5 [×2], C202D4, Dic5⋊D4 [×2], C5×C4⋊D4, C10.1152+ 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.47C24, D42D5 [×4], C23×D5, C2×D42D5 [×2], D48D10, C10.1152+ 1+4

Smallest permutation representation of C10.1152+ 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 50 21 33)(2 41 22 34)(3 42 23 35)(4 43 24 36)(5 44 25 37)(6 45 26 38)(7 46 27 39)(8 47 28 40)(9 48 29 31)(10 49 30 32)(11 138 158 145)(12 139 159 146)(13 140 160 147)(14 131 151 148)(15 132 152 149)(16 133 153 150)(17 134 154 141)(18 135 155 142)(19 136 156 143)(20 137 157 144)(51 88 68 71)(52 89 69 72)(53 90 70 73)(54 81 61 74)(55 82 62 75)(56 83 63 76)(57 84 64 77)(58 85 65 78)(59 86 66 79)(60 87 67 80)(91 123 108 116)(92 124 109 117)(93 125 110 118)(94 126 101 119)(95 127 102 120)(96 128 103 111)(97 129 104 112)(98 130 105 113)(99 121 106 114)(100 122 107 115)
(1 133 6 138)(2 132 7 137)(3 131 8 136)(4 140 9 135)(5 139 10 134)(11 50 16 45)(12 49 17 44)(13 48 18 43)(14 47 19 42)(15 46 20 41)(21 150 26 145)(22 149 27 144)(23 148 28 143)(24 147 29 142)(25 146 30 141)(31 155 36 160)(32 154 37 159)(33 153 38 158)(34 152 39 157)(35 151 40 156)(51 95 56 100)(52 94 57 99)(53 93 58 98)(54 92 59 97)(55 91 60 96)(61 109 66 104)(62 108 67 103)(63 107 68 102)(64 106 69 101)(65 105 70 110)(71 127 76 122)(72 126 77 121)(73 125 78 130)(74 124 79 129)(75 123 80 128)(81 117 86 112)(82 116 87 111)(83 115 88 120)(84 114 89 119)(85 113 90 118)
(1 105 21 98)(2 104 22 97)(3 103 23 96)(4 102 24 95)(5 101 25 94)(6 110 26 93)(7 109 27 92)(8 108 28 91)(9 107 29 100)(10 106 30 99)(11 90 158 73)(12 89 159 72)(13 88 160 71)(14 87 151 80)(15 86 152 79)(16 85 153 78)(17 84 154 77)(18 83 155 76)(19 82 156 75)(20 81 157 74)(31 127 48 120)(32 126 49 119)(33 125 50 118)(34 124 41 117)(35 123 42 116)(36 122 43 115)(37 121 44 114)(38 130 45 113)(39 129 46 112)(40 128 47 111)(51 135 68 142)(52 134 69 141)(53 133 70 150)(54 132 61 149)(55 131 62 148)(56 140 63 147)(57 139 64 146)(58 138 65 145)(59 137 66 144)(60 136 67 143)
(1 73 6 78)(2 72 7 77)(3 71 8 76)(4 80 9 75)(5 79 10 74)(11 98 16 93)(12 97 17 92)(13 96 18 91)(14 95 19 100)(15 94 20 99)(21 90 26 85)(22 89 27 84)(23 88 28 83)(24 87 29 82)(25 86 30 81)(31 62 36 67)(32 61 37 66)(33 70 38 65)(34 69 39 64)(35 68 40 63)(41 52 46 57)(42 51 47 56)(43 60 48 55)(44 59 49 54)(45 58 50 53)(101 157 106 152)(102 156 107 151)(103 155 108 160)(104 154 109 159)(105 153 110 158)(111 142 116 147)(112 141 117 146)(113 150 118 145)(114 149 119 144)(115 148 120 143)(121 132 126 137)(122 131 127 136)(123 140 128 135)(124 139 129 134)(125 138 130 133)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F···4M4N4O4P5A5B10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order122222222444444···44445510···10101010101010101020···2020202020
size11112244202244410···10202020222···2444488884···48888

53 irreducible representations

dim1111111111122222224444
type+++++++++++++++++--+
imageC1C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4D10D10D10D102+ 1+4D42D5D42D5D48D10
kernelC10.1152+ 1+4C23.D10C22.D20C4.Dic10C4⋊C47D5C2×C4⋊Dic5C4×C5⋊D4D4×Dic5C202D4Dic5⋊D4C5×C4⋊D4C4⋊D4C20C2×C10C22⋊C4C4⋊C4C22×C4C2×D4C10C4C22C2
# reps1221111312124442261444

Matrix representation of C10.1152+ 1+4 in GL6(𝔽41)

4000000
0400000
006700
0035000
0000400
0000040
,
3200000
3290000
001000
000100
00004032
000001
,
1390000
1400000
006700
00363500
0000320
0000032
,
4020000
4010000
006700
00363500
000010
00001840
,
3200000
0320000
006700
00363500
0000940
0000032

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,7,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,32,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,32,1],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,6,36,0,0,0,0,7,35,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,6,36,0,0,0,0,7,35,0,0,0,0,0,0,1,18,0,0,0,0,0,40],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,6,36,0,0,0,0,7,35,0,0,0,0,0,0,9,0,0,0,0,0,40,32] >;

C10.1152+ 1+4 in GAP, Magma, Sage, TeX

C_{10}._{115}2_+^{1+4}
% in TeX

G:=Group("C10.115ES+(2,2)");
// GroupNames label

G:=SmallGroup(320,1290);
// by ID

G=gap.SmallGroup(320,1290);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,675,570,185,136,12550]);
// Polycyclic

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

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