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G = C10.232- 1+4order 320 = 26·5

23rd non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.232- 1+4, C20⋊Q827C2, C22⋊Q818D5, C4⋊C4.196D10, (Q8×Dic5)⋊15C2, D103Q822C2, (C2×C20).62C23, (C2×Q8).131D10, C22⋊C4.21D10, Dic53Q829C2, C20.212(C4○D4), (C2×C10).185C24, (C22×C4).247D10, C4.101(D42D5), Dic5.63(C4○D4), C23.D1025C2, C4⋊Dic5.376C22, Dic5.5D4.2C2, (Q8×C10).115C22, (C22×D5).76C23, C22.206(C23×D5), C23.124(C22×D5), C23.21D1030C2, (C22×C20).260C22, (C22×C10).213C23, C55(C22.50C24), (C4×Dic5).121C22, (C2×Dic5).249C23, C10.D4.33C22, C23.D5.124C22, D10⋊C4.130C22, C2.24(Q8.10D10), (C2×Dic10).169C22, C2.56(D5×C4○D4), C4⋊C4⋊D520C2, C4⋊C47D529C2, (C4×C5⋊D4).10C2, (C5×C22⋊Q8)⋊21C2, C10.168(C2×C4○D4), C2.48(C2×D42D5), (C2×C4×D5).111C22, (C2×C4).55(C22×D5), (C5×C4⋊C4).166C22, (C2×C5⋊D4).140C22, (C5×C22⋊C4).40C22, SmallGroup(320,1313)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.232- 1+4
C1C5C10C2×C10C22×D5C2×C4×D5D103Q8 — C10.232- 1+4
C5C2×C10 — C10.232- 1+4
C1C22C22⋊Q8

Generators and relations for C10.232- 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, bd=db, be=eb, dcd-1=a5c, ce=ec, ede-1=b2d >

Subgroups: 646 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], Q8 [×6], C23, C23, D5, C10 [×3], C10, C42 [×7], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×5], D10 [×3], C2×C10, C2×C10 [×3], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8, C22⋊Q8, C4.4D4 [×2], C422C2 [×4], C4⋊Q8, Dic10 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×C10, C22.50C24, C4×Dic5, C4×Dic5 [×6], C10.D4, C10.D4 [×4], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×4], C23.D5, C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C5⋊D4, C22×C20, Q8×C10, C23.D10 [×2], Dic5.5D4 [×2], Dic53Q8 [×2], C20⋊Q8, C4⋊C47D5, C4⋊C4⋊D5 [×2], C23.21D10, C4×C5⋊D4, Q8×Dic5, D103Q8, C5×C22⋊Q8, C10.232- 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], C22.50C24, D42D5 [×2], C23×D5, C2×D42D5, Q8.10D10, D5×C4○D4, C10.232- 1+4

Smallest permutation representation of C10.232- 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)
(1 118 13 108)(2 119 14 109)(3 120 15 110)(4 111 16 101)(5 112 17 102)(6 113 18 103)(7 114 19 104)(8 115 20 105)(9 116 11 106)(10 117 12 107)(21 96 31 86)(22 97 32 87)(23 98 33 88)(24 99 34 89)(25 100 35 90)(26 91 36 81)(27 92 37 82)(28 93 38 83)(29 94 39 84)(30 95 40 85)(41 151 51 141)(42 152 52 142)(43 153 53 143)(44 154 54 144)(45 155 55 145)(46 156 56 146)(47 157 57 147)(48 158 58 148)(49 159 59 149)(50 160 60 150)(61 131 71 121)(62 132 72 122)(63 133 73 123)(64 134 74 124)(65 135 75 125)(66 136 76 126)(67 137 77 127)(68 138 78 128)(69 139 79 129)(70 140 80 130)
(1 108 6 103)(2 107 7 102)(3 106 8 101)(4 105 9 110)(5 104 10 109)(11 120 16 115)(12 119 17 114)(13 118 18 113)(14 117 19 112)(15 116 20 111)(21 95 26 100)(22 94 27 99)(23 93 28 98)(24 92 29 97)(25 91 30 96)(31 85 36 90)(32 84 37 89)(33 83 38 88)(34 82 39 87)(35 81 40 86)(41 150 46 145)(42 149 47 144)(43 148 48 143)(44 147 49 142)(45 146 50 141)(51 160 56 155)(52 159 57 154)(53 158 58 153)(54 157 59 152)(55 156 60 151)(61 135 66 140)(62 134 67 139)(63 133 68 138)(64 132 69 137)(65 131 70 136)(71 125 76 130)(72 124 77 129)(73 123 78 128)(74 122 79 127)(75 121 80 126)
(1 48 18 53)(2 47 19 52)(3 46 20 51)(4 45 11 60)(5 44 12 59)(6 43 13 58)(7 42 14 57)(8 41 15 56)(9 50 16 55)(10 49 17 54)(21 75 36 70)(22 74 37 69)(23 73 38 68)(24 72 39 67)(25 71 40 66)(26 80 31 65)(27 79 32 64)(28 78 33 63)(29 77 34 62)(30 76 35 61)(81 140 96 125)(82 139 97 124)(83 138 98 123)(84 137 99 122)(85 136 100 121)(86 135 91 130)(87 134 92 129)(88 133 93 128)(89 132 94 127)(90 131 95 126)(101 145 116 160)(102 144 117 159)(103 143 118 158)(104 142 119 157)(105 141 120 156)(106 150 111 155)(107 149 112 154)(108 148 113 153)(109 147 114 152)(110 146 115 151)
(1 93 13 83)(2 94 14 84)(3 95 15 85)(4 96 16 86)(5 97 17 87)(6 98 18 88)(7 99 19 89)(8 100 20 90)(9 91 11 81)(10 92 12 82)(21 111 31 101)(22 112 32 102)(23 113 33 103)(24 114 34 104)(25 115 35 105)(26 116 36 106)(27 117 37 107)(28 118 38 108)(29 119 39 109)(30 120 40 110)(41 131 51 121)(42 132 52 122)(43 133 53 123)(44 134 54 124)(45 135 55 125)(46 136 56 126)(47 137 57 127)(48 138 58 128)(49 139 59 129)(50 140 60 130)(61 151 71 141)(62 152 72 142)(63 153 73 143)(64 154 74 144)(65 155 75 145)(66 156 76 146)(67 157 77 147)(68 158 78 148)(69 159 79 149)(70 160 80 150)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4P4Q4R4S5A5B10A···10F10G10H10I10J20A···20H20I···20P
order122222444444444···44445510···101010101020···2020···20
size11114202222444410···10202020222···244444···48···8

53 irreducible representations

dim11111111111122222224444
type+++++++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4D10D10D10D102- 1+4D42D5Q8.10D10D5×C4○D4
kernelC10.232- 1+4C23.D10Dic5.5D4Dic53Q8C20⋊Q8C4⋊C47D5C4⋊C4⋊D5C23.21D10C4×C5⋊D4Q8×Dic5D103Q8C5×C22⋊Q8C22⋊Q8Dic5C20C22⋊C4C4⋊C4C22×C4C2×Q8C10C4C2C2
# reps12221121111124446221444

Matrix representation of C10.232- 1+4 in GL6(𝔽41)

4000000
0400000
00343500
007000
0000400
0000040
,
4010000
010000
001000
000100
0000320
0000032
,
1400000
2400000
00343500
008700
000090
00003232
,
3290000
090000
007600
00333400
000012
0000040
,
100000
010000
0040000
0004000
000090
00003232

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,35,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,2,0,0,0,0,40,40,0,0,0,0,0,0,34,8,0,0,0,0,35,7,0,0,0,0,0,0,9,32,0,0,0,0,0,32],[32,0,0,0,0,0,9,9,0,0,0,0,0,0,7,33,0,0,0,0,6,34,0,0,0,0,0,0,1,0,0,0,0,0,2,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,32,0,0,0,0,0,32] >;

C10.232- 1+4 in GAP, Magma, Sage, TeX

C_{10}._{23}2_-^{1+4}
% in TeX

G:=Group("C10.23ES-(2,2)");
// GroupNames label

G:=SmallGroup(320,1313);
// by ID

G=gap.SmallGroup(320,1313);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,100,1571,297,12550]);
// 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,b*d=d*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

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