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G = C10.642+ 1+4order 320 = 26·5

64th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.642+ 1+4, C202D431C2, C4⋊C4.198D10, D208C434C2, D10⋊D433C2, (C2×D4).101D10, C22⋊C4.68D10, (C22×C4).50D10, D10.54(C4○D4), Dic5⋊D422C2, Dic54D423C2, (C2×C20).600C23, (C2×C10).205C24, C22.D410D5, D10.12D434C2, C2.66(D46D10), C23.29(C22×D5), Dic5.42(C4○D4), Dic5.Q828C2, (D4×C10).143C22, (C2×D20).163C22, C23.D1033C2, C4⋊Dic5.229C22, (C22×C10).37C23, C22.226(C23×D5), C23.11D1014C2, (C22×C20).369C22, C59(C22.47C24), (C4×Dic5).232C22, (C2×Dic5).256C23, C10.D4.43C22, (C22×D5).221C23, C23.D5.127C22, D10⋊C4.132C22, (C22×Dic5).131C22, (D5×C4⋊C4)⋊34C2, (C4×C5⋊D4)⋊49C2, C2.67(D5×C4○D4), C4⋊C4⋊D529C2, C10.179(C2×C4○D4), (C2×C4×D5).123C22, (C2×C4).67(C22×D5), (C5×C4⋊C4).178C22, (C2×C5⋊D4).49C22, (C5×C22.D4)⋊13C2, (C5×C22⋊C4).53C22, SmallGroup(320,1333)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.642+ 1+4
C1C5C10C2×C10C22×D5C2×C4×D5D5×C4⋊C4 — C10.642+ 1+4
C5C2×C10 — C10.642+ 1+4
C1C22C22.D4

Generators and relations for C10.642+ 1+4
 G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=a5b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, dbd-1=ebe=a5b, cd=dc, ce=ec, ede=a5b2d >

Subgroups: 854 in 238 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C4 [×12], C22, C22 [×13], C5, C2×C4 [×5], C2×C4 [×14], D4 [×10], C23 [×2], C23 [×2], D5 [×3], C10 [×3], C10 [×2], C42 [×3], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×5], Dic5 [×2], Dic5 [×5], C20 [×5], D10 [×2], D10 [×5], C2×C10, C2×C10 [×6], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4, C22.D4, C42.C2, C422C2 [×2], C4×D5 [×5], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×7], C2×C20 [×5], C2×C20, C5×D4, C22×D5 [×2], C22×C10 [×2], C22.47C24, C4×Dic5 [×3], C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×3], C5×C22⋊C4 [×3], C5×C4⋊C4 [×2], C2×C4×D5 [×4], C2×D20, C22×Dic5, C2×C5⋊D4 [×4], C22×C20, D4×C10, C23.11D10, C23.D10, Dic54D4, D10.12D4, D10⋊D4 [×2], Dic5.Q8, D5×C4⋊C4, D208C4, C4⋊C4⋊D5, C4×C5⋊D4 [×2], C202D4, Dic5⋊D4, C5×C22.D4, C10.642+ 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, C23×D5, D46D10, D5×C4○D4 [×2], C10.642+ 1+4

Smallest permutation representation of C10.642+ 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)
(1 138 13 128)(2 139 14 129)(3 140 15 130)(4 131 16 121)(5 132 17 122)(6 133 18 123)(7 134 19 124)(8 135 20 125)(9 136 11 126)(10 137 12 127)(21 156 31 146)(22 157 32 147)(23 158 33 148)(24 159 34 149)(25 160 35 150)(26 151 36 141)(27 152 37 142)(28 153 38 143)(29 154 39 144)(30 155 40 145)(41 91 51 81)(42 92 52 82)(43 93 53 83)(44 94 54 84)(45 95 55 85)(46 96 56 86)(47 97 57 87)(48 98 58 88)(49 99 59 89)(50 100 60 90)(61 111 71 101)(62 112 72 102)(63 113 73 103)(64 114 74 104)(65 115 75 105)(66 116 76 106)(67 117 77 107)(68 118 78 108)(69 119 79 109)(70 120 80 110)
(1 88 6 83)(2 89 7 84)(3 90 8 85)(4 81 9 86)(5 82 10 87)(11 96 16 91)(12 97 17 92)(13 98 18 93)(14 99 19 94)(15 100 20 95)(21 106 26 101)(22 107 27 102)(23 108 28 103)(24 109 29 104)(25 110 30 105)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)
(1 133 18 128)(2 132 19 127)(3 131 20 126)(4 140 11 125)(5 139 12 124)(6 138 13 123)(7 137 14 122)(8 136 15 121)(9 135 16 130)(10 134 17 129)(21 145 36 160)(22 144 37 159)(23 143 38 158)(24 142 39 157)(25 141 40 156)(26 150 31 155)(27 149 32 154)(28 148 33 153)(29 147 34 152)(30 146 35 151)(41 85 56 100)(42 84 57 99)(43 83 58 98)(44 82 59 97)(45 81 60 96)(46 90 51 95)(47 89 52 94)(48 88 53 93)(49 87 54 92)(50 86 55 91)(61 115 76 110)(62 114 77 109)(63 113 78 108)(64 112 79 107)(65 111 80 106)(66 120 71 105)(67 119 72 104)(68 118 73 103)(69 117 74 102)(70 116 75 101)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 146)(122 147)(123 148)(124 149)(125 150)(126 141)(127 142)(128 143)(129 144)(130 145)(131 156)(132 157)(133 158)(134 159)(135 160)(136 151)(137 152)(138 153)(139 154)(140 155)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H···4M4N4O4P5A5B10A···10F10G10H10I10J10K10L20A···20H20I···20N
order12222222244444444···44445510···1010101010101020···2020···20
size111144101020222244410···10202020222···24444884···48···8

53 irreducible representations

dim111111111111112222222444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4D10D10D10D102+ 1+4D46D10D5×C4○D4
kernelC10.642+ 1+4C23.11D10C23.D10Dic54D4D10.12D4D10⋊D4Dic5.Q8D5×C4⋊C4D208C4C4⋊C4⋊D5C4×C5⋊D4C202D4Dic5⋊D4C5×C22.D4C22.D4Dic5D10C22⋊C4C4⋊C4C22×C4C2×D4C10C2C2
# reps111112111121112446422148

Matrix representation of C10.642+ 1+4 in GL6(𝔽41)

4000000
0400000
0040000
0004000
00003535
0000640
,
9250000
5320000
00404000
002100
000010
000001
,
3200000
0320000
009000
00233200
0000400
0000040
,
9250000
0320000
001000
00394000
000010
0000640
,
100000
37400000
001000
00394000
000010
000001

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40],[9,5,0,0,0,0,25,32,0,0,0,0,0,0,40,2,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,23,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,25,32,0,0,0,0,0,0,1,39,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[1,37,0,0,0,0,0,40,0,0,0,0,0,0,1,39,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C10.642+ 1+4 in GAP, Magma, Sage, TeX

C_{10}._{64}2_+^{1+4}
% in TeX

G:=Group("C10.64ES+(2,2)");
// GroupNames label

G:=SmallGroup(320,1333);
// by ID

G=gap.SmallGroup(320,1333);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,219,184,1571,297,12550]);
// Polycyclic

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

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