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G = Q8⋊C4⋊D5order 320 = 26·5

15th semidirect product of Q8⋊C4 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20⋊Q85C2, C4⋊C4.18D10, C408C419C2, Q8⋊C415D5, Q8⋊Dic51C2, (C2×C8).173D10, (C2×Q8).11D10, D205C4.7C2, D206C4.1C2, C20.15(C4○D4), C4.29(C4○D20), (C2×Dic5).34D4, C22.189(D4×D5), C4.55(D42D5), C2.15(D40⋊C2), C10.60(C8⋊C22), (C2×C40).193C22, (C2×C20).235C23, C2.9(Q16⋊D5), C20.23D4.4C2, (C2×D20).61C22, C4⋊Dic5.84C22, (Q8×C10).18C22, C10.27(C4.4D4), C10.54(C8.C22), (C4×Dic5).25C22, C53(C42.28C22), C2.17(Dic5.5D4), (C5×Q8⋊C4)⋊16C2, (C2×C10).248(C2×D4), (C5×C4⋊C4).36C22, (C2×C52C8).30C22, (C2×C4).342(C22×D5), SmallGroup(320,422)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Q8⋊C4⋊D5
C1C5C10C20C2×C20C4×Dic5C20⋊Q8 — Q8⋊C4⋊D5
C5C10C2×C20 — Q8⋊C4⋊D5
C1C22C2×C4Q8⋊C4

Generators and relations for Q8⋊C4⋊D5
 G = < a,b,c,d,e | a4=c4=d5=e2=1, b2=a2, bab-1=cac-1=eae=a-1, ad=da, cbc-1=a-1b, bd=db, ebe=a2bc2, cd=dc, ece=ac-1, ede=d-1 >

Subgroups: 462 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×4], C23, D5, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, C2×D4, C2×Q8, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C8⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C52C8, C40, Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C42.28C22, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×D20, Q8×C10, D206C4, C408C4, D205C4, Q8⋊Dic5, C5×Q8⋊C4, C20⋊Q8, C20.23D4, Q8⋊C4⋊D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8⋊C22, C8.C22, C22×D5, C42.28C22, C4○D20, D4×D5, D42D5, Dic5.5D4, D40⋊C2, Q16⋊D5, Q8⋊C4⋊D5

Smallest permutation representation of Q8⋊C4⋊D5
On 160 points
Generators in S160
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 80)(81 101 86 106)(82 102 87 107)(83 103 88 108)(84 104 89 109)(85 105 90 110)(91 111 96 116)(92 112 97 117)(93 113 98 118)(94 114 99 119)(95 115 100 120)(121 146 126 141)(122 147 127 142)(123 148 128 143)(124 149 129 144)(125 150 130 145)(131 156 136 151)(132 157 137 152)(133 158 138 153)(134 159 139 154)(135 160 140 155)
(1 59 9 54)(2 60 10 55)(3 56 6 51)(4 57 7 52)(5 58 8 53)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 76 26 71)(22 77 27 72)(23 78 28 73)(24 79 29 74)(25 80 30 75)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)(81 156 86 151)(82 157 87 152)(83 158 88 153)(84 159 89 154)(85 160 90 155)(91 146 96 141)(92 147 97 142)(93 148 98 143)(94 149 99 144)(95 150 100 145)(101 131 106 136)(102 132 107 137)(103 133 108 138)(104 134 109 139)(105 135 110 140)(111 121 116 126)(112 122 117 127)(113 123 118 128)(114 124 119 129)(115 125 120 130)
(1 94 14 84)(2 95 15 85)(3 91 11 81)(4 92 12 82)(5 93 13 83)(6 96 16 86)(7 97 17 87)(8 98 18 88)(9 99 19 89)(10 100 20 90)(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)
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 56)(42 60)(43 59)(44 58)(45 57)(46 51)(47 55)(48 54)(49 53)(50 52)(61 71)(62 75)(63 74)(64 73)(65 72)(66 76)(67 80)(68 79)(69 78)(70 77)(81 116)(82 120)(83 119)(84 118)(85 117)(86 111)(87 115)(88 114)(89 113)(90 112)(91 106)(92 110)(93 109)(94 108)(95 107)(96 101)(97 105)(98 104)(99 103)(100 102)(121 146)(122 150)(123 149)(124 148)(125 147)(126 141)(127 145)(128 144)(129 143)(130 142)(131 156)(132 160)(133 159)(134 158)(135 157)(136 151)(137 155)(138 154)(139 153)(140 152)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222444444455888810···102020202020···2040···40
size1111402288202040224420202···244448···84···4

44 irreducible representations

dim111111112222222444444
type++++++++++++++--++
imageC1C2C2C2C2C2C2C2D4D5C4○D4D10D10D10C4○D20C8⋊C22C8.C22D42D5D4×D5D40⋊C2Q16⋊D5
kernelQ8⋊C4⋊D5D206C4C408C4D205C4Q8⋊Dic5C5×Q8⋊C4C20⋊Q8C20.23D4C2×Dic5Q8⋊C4C20C4⋊C4C2×C8C2×Q8C4C10C10C4C22C2C2
# reps111111112242228112244

Matrix representation of Q8⋊C4⋊D5 in GL8(𝔽41)

400000000
040000000
004000000
000400000
00000100
000040000
000000040
00000010
,
2413180000
40175180000
0018400000
0036230000
00000010
00000001
000040000
000004000
,
3101530000
0310380000
37371000000
0200100000
00003383011
0000883030
000030303333
00001130338
,
01000000
4034000000
004010000
005350000
00001000
00000100
00000010
00000001
,
01000000
10000000
00100000
0036400000
00001000
000004000
000000400
00000001

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0],[24,40,0,0,0,0,0,0,1,17,0,0,0,0,0,0,3,5,18,36,0,0,0,0,18,18,40,23,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[31,0,37,0,0,0,0,0,0,31,37,20,0,0,0,0,15,0,10,0,0,0,0,0,3,38,0,10,0,0,0,0,0,0,0,0,33,8,30,11,0,0,0,0,8,8,30,30,0,0,0,0,30,30,33,33,0,0,0,0,11,30,33,8],[0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,40,5,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1] >;

Q8⋊C4⋊D5 in GAP, Magma, Sage, TeX

Q_8\rtimes C_4\rtimes D_5
% in TeX

G:=Group("Q8:C4:D5");
// GroupNames label

G:=SmallGroup(320,422);
// by ID

G=gap.SmallGroup(320,422);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,1094,135,184,570,297,136,12550]);
// Polycyclic

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

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