direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.Dic5, C20.75C24, C10⋊5(C8○D4), C4○D4.44D10, (D4×C10).25C4, (Q8×C10).22C4, C4○D4.4Dic5, Q8.9(C2×Dic5), D4.8(C2×Dic5), C4.74(C23×D5), C10.69(C23×C4), C5⋊2C8.44C23, (C2×Q8).10Dic5, (C2×D4).12Dic5, (C2×C20).553C23, C20.156(C22×C4), (C22×C4).386D10, C4.Dic5⋊34C22, C23.19(C2×Dic5), C2.10(C23×Dic5), C4.40(C22×Dic5), (C22×C20).288C22, C22.33(C22×Dic5), C5⋊7(C2×C8○D4), (C5×C4○D4).7C4, (C10×C4○D4).9C2, (C2×C4○D4).13D5, (C5×D4).39(C2×C4), (C5×Q8).42(C2×C4), (C2×C20).308(C2×C4), (C22×C5⋊2C8)⋊14C2, (C2×C5⋊2C8)⋊41C22, (C2×C4.Dic5)⋊28C2, (C2×C4).56(C2×Dic5), (C5×C4○D4).48C22, (C2×C4).831(C22×D5), (C22×C10).148(C2×C4), (C2×C10).129(C22×C4), SmallGroup(320,1490)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8.Dic5
G = < a,b,c,d,e | a2=b4=1, c2=d10=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >
Subgroups: 494 in 266 conjugacy classes, 191 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C5, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×2], C10 [×6], C2×C8 [×16], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×2], C20 [×6], C2×C10, C2×C10 [×6], C2×C10 [×6], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C5⋊2C8 [×8], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C2×C8○D4, C2×C5⋊2C8, C2×C5⋊2C8 [×15], C4.Dic5 [×12], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C22×C5⋊2C8 [×3], C2×C4.Dic5 [×3], Q8.Dic5 [×8], C10×C4○D4, C2×Q8.Dic5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C8○D4 [×2], C23×C4, C2×Dic5 [×28], C22×D5 [×7], C2×C8○D4, C22×Dic5 [×14], C23×D5, Q8.Dic5 [×2], C23×Dic5, C2×Q8.Dic5
►On 160 points
Generators in S160
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 76)(42 77)(43 78)(44 79)(45 80)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)(89 113)(90 114)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)(97 101)(98 102)(99 103)(100 104)(121 158)(122 159)(123 160)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(131 148)(132 149)(133 150)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 157)
(1 56 11 46)(2 57 12 47)(3 58 13 48)(4 59 14 49)(5 60 15 50)(6 41 16 51)(7 42 17 52)(8 43 18 53)(9 44 19 54)(10 45 20 55)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 132 91 122)(82 133 92 123)(83 134 93 124)(84 135 94 125)(85 136 95 126)(86 137 96 127)(87 138 97 128)(88 139 98 129)(89 140 99 130)(90 121 100 131)(101 145 111 155)(102 146 112 156)(103 147 113 157)(104 148 114 158)(105 149 115 159)(106 150 116 160)(107 151 117 141)(108 152 118 142)(109 153 119 143)(110 154 120 144)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)(121 136 131 126)(122 137 132 127)(123 138 133 128)(124 139 134 129)(125 140 135 130)(141 156 151 146)(142 157 152 147)(143 158 153 148)(144 159 154 149)(145 160 155 150)
(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)
(1 101 6 106 11 111 16 116)(2 110 7 115 12 120 17 105)(3 119 8 104 13 109 18 114)(4 108 9 113 14 118 19 103)(5 117 10 102 15 107 20 112)(21 97 26 82 31 87 36 92)(22 86 27 91 32 96 37 81)(23 95 28 100 33 85 38 90)(24 84 29 89 34 94 39 99)(25 93 30 98 35 83 40 88)(41 150 46 155 51 160 56 145)(42 159 47 144 52 149 57 154)(43 148 48 153 53 158 58 143)(44 157 49 142 54 147 59 152)(45 146 50 151 55 156 60 141)(61 138 66 123 71 128 76 133)(62 127 67 132 72 137 77 122)(63 136 68 121 73 126 78 131)(64 125 69 130 74 135 79 140)(65 134 70 139 75 124 80 129)
G:=sub<Sym(160)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,76)(42,77)(43,78)(44,79)(45,80)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(97,101)(98,102)(99,103)(100,104)(121,158)(122,159)(123,160)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157), (1,56,11,46)(2,57,12,47)(3,58,13,48)(4,59,14,49)(5,60,15,50)(6,41,16,51)(7,42,17,52)(8,43,18,53)(9,44,19,54)(10,45,20,55)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)(81,132,91,122)(82,133,92,123)(83,134,93,124)(84,135,94,125)(85,136,95,126)(86,137,96,127)(87,138,97,128)(88,139,98,129)(89,140,99,130)(90,121,100,131)(101,145,111,155)(102,146,112,156)(103,147,113,157)(104,148,114,158)(105,149,115,159)(106,150,116,160)(107,151,117,141)(108,152,118,142)(109,153,119,143)(110,154,120,144), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120)(121,136,131,126)(122,137,132,127)(123,138,133,128)(124,139,134,129)(125,140,135,130)(141,156,151,146)(142,157,152,147)(143,158,153,148)(144,159,154,149)(145,160,155,150), (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), (1,101,6,106,11,111,16,116)(2,110,7,115,12,120,17,105)(3,119,8,104,13,109,18,114)(4,108,9,113,14,118,19,103)(5,117,10,102,15,107,20,112)(21,97,26,82,31,87,36,92)(22,86,27,91,32,96,37,81)(23,95,28,100,33,85,38,90)(24,84,29,89,34,94,39,99)(25,93,30,98,35,83,40,88)(41,150,46,155,51,160,56,145)(42,159,47,144,52,149,57,154)(43,148,48,153,53,158,58,143)(44,157,49,142,54,147,59,152)(45,146,50,151,55,156,60,141)(61,138,66,123,71,128,76,133)(62,127,67,132,72,137,77,122)(63,136,68,121,73,126,78,131)(64,125,69,130,74,135,79,140)(65,134,70,139,75,124,80,129)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,76)(42,77)(43,78)(44,79)(45,80)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(97,101)(98,102)(99,103)(100,104)(121,158)(122,159)(123,160)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157), (1,56,11,46)(2,57,12,47)(3,58,13,48)(4,59,14,49)(5,60,15,50)(6,41,16,51)(7,42,17,52)(8,43,18,53)(9,44,19,54)(10,45,20,55)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)(81,132,91,122)(82,133,92,123)(83,134,93,124)(84,135,94,125)(85,136,95,126)(86,137,96,127)(87,138,97,128)(88,139,98,129)(89,140,99,130)(90,121,100,131)(101,145,111,155)(102,146,112,156)(103,147,113,157)(104,148,114,158)(105,149,115,159)(106,150,116,160)(107,151,117,141)(108,152,118,142)(109,153,119,143)(110,154,120,144), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120)(121,136,131,126)(122,137,132,127)(123,138,133,128)(124,139,134,129)(125,140,135,130)(141,156,151,146)(142,157,152,147)(143,158,153,148)(144,159,154,149)(145,160,155,150), (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), (1,101,6,106,11,111,16,116)(2,110,7,115,12,120,17,105)(3,119,8,104,13,109,18,114)(4,108,9,113,14,118,19,103)(5,117,10,102,15,107,20,112)(21,97,26,82,31,87,36,92)(22,86,27,91,32,96,37,81)(23,95,28,100,33,85,38,90)(24,84,29,89,34,94,39,99)(25,93,30,98,35,83,40,88)(41,150,46,155,51,160,56,145)(42,159,47,144,52,149,57,154)(43,148,48,153,53,158,58,143)(44,157,49,142,54,147,59,152)(45,146,50,151,55,156,60,141)(61,138,66,123,71,128,76,133)(62,127,67,132,72,137,77,122)(63,136,68,121,73,126,78,131)(64,125,69,130,74,135,79,140)(65,134,70,139,75,124,80,129) );
G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,76),(42,77),(43,78),(44,79),(45,80),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72),(58,73),(59,74),(60,75),(81,105),(82,106),(83,107),(84,108),(85,109),(86,110),(87,111),(88,112),(89,113),(90,114),(91,115),(92,116),(93,117),(94,118),(95,119),(96,120),(97,101),(98,102),(99,103),(100,104),(121,158),(122,159),(123,160),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146),(130,147),(131,148),(132,149),(133,150),(134,151),(135,152),(136,153),(137,154),(138,155),(139,156),(140,157)], [(1,56,11,46),(2,57,12,47),(3,58,13,48),(4,59,14,49),(5,60,15,50),(6,41,16,51),(7,42,17,52),(8,43,18,53),(9,44,19,54),(10,45,20,55),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(26,76,36,66),(27,77,37,67),(28,78,38,68),(29,79,39,69),(30,80,40,70),(81,132,91,122),(82,133,92,123),(83,134,93,124),(84,135,94,125),(85,136,95,126),(86,137,96,127),(87,138,97,128),(88,139,98,129),(89,140,99,130),(90,121,100,131),(101,145,111,155),(102,146,112,156),(103,147,113,157),(104,148,114,158),(105,149,115,159),(106,150,116,160),(107,151,117,141),(108,152,118,142),(109,153,119,143),(110,154,120,144)], [(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,26,31,36),(22,27,32,37),(23,28,33,38),(24,29,34,39),(25,30,35,40),(41,56,51,46),(42,57,52,47),(43,58,53,48),(44,59,54,49),(45,60,55,50),(61,76,71,66),(62,77,72,67),(63,78,73,68),(64,79,74,69),(65,80,75,70),(81,86,91,96),(82,87,92,97),(83,88,93,98),(84,89,94,99),(85,90,95,100),(101,106,111,116),(102,107,112,117),(103,108,113,118),(104,109,114,119),(105,110,115,120),(121,136,131,126),(122,137,132,127),(123,138,133,128),(124,139,134,129),(125,140,135,130),(141,156,151,146),(142,157,152,147),(143,158,153,148),(144,159,154,149),(145,160,155,150)], [(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)], [(1,101,6,106,11,111,16,116),(2,110,7,115,12,120,17,105),(3,119,8,104,13,109,18,114),(4,108,9,113,14,118,19,103),(5,117,10,102,15,107,20,112),(21,97,26,82,31,87,36,92),(22,86,27,91,32,96,37,81),(23,95,28,100,33,85,38,90),(24,84,29,89,34,94,39,99),(25,93,30,98,35,83,40,88),(41,150,46,155,51,160,56,145),(42,159,47,144,52,149,57,154),(43,148,48,153,53,158,58,143),(44,157,49,142,54,147,59,152),(45,146,50,151,55,156,60,141),(61,138,66,123,71,128,76,133),(62,127,67,132,72,137,77,122),(63,136,68,121,73,126,78,131),(64,125,69,130,74,135,79,140),(65,134,70,139,75,124,80,129)])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8T | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D5 | D10 | Dic5 | Dic5 | Dic5 | D10 | C8○D4 | Q8.Dic5 |
| kernel | C2×Q8.Dic5 | C22×C5⋊2C8 | C2×C4.Dic5 | Q8.Dic5 | C10×C4○D4 | D4×C10 | Q8×C10 | C5×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C4○D4 | C10 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 6 | 2 | 8 | 2 | 6 | 6 | 2 | 8 | 8 | 8 | 8 |
Matrix representation of C2×Q8.Dic5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 9 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 40 | 40 | 0 | 0 |
| 8 | 7 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 9 | 25 | 0 | 0 |
| 5 | 32 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,0,9,0,0,9,0],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,32],[40,8,0,0,40,7,0,0,0,0,9,0,0,0,0,9],[9,5,0,0,25,32,0,0,0,0,3,0,0,0,0,3] >;
C2×Q8.Dic5 in GAP, Magma, Sage, TeX
C_2\times Q_8.{\rm Dic}_5 % in TeX
G:=Group("C2xQ8.Dic5"); // GroupNames label
G:=SmallGroup(320,1490);
// by ID
G=gap.SmallGroup(320,1490);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,297,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^10=b^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^9>;
// generators/relations