direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C22⋊Q8, C23⋊3(C5×Q8), C4.63(D4×C10), C22⋊1(Q8×C10), (C22×C10)⋊6Q8, C20.470(C2×D4), (C2×C20).524D4, (C22×Q8)⋊3C10, C24.31(C2×C10), (C23×C20).25C2, (C23×C4).10C10, (Q8×C10)⋊48C22, C22.60(D4×C10), C10.57(C22×Q8), (C2×C10).343C24, (C2×C20).656C23, C10.182(C22×D4), (C23×C10).91C22, C23.70(C22×C10), C22.17(C23×C10), (C22×C20).444C22, (C22×C10).258C23, C2.6(D4×C2×C10), C2.3(Q8×C2×C10), (C10×C4⋊C4)⋊42C2, (C2×C4⋊C4)⋊15C10, (Q8×C2×C10)⋊15C2, (C2×C10)⋊5(C2×Q8), C4⋊C4⋊10(C2×C10), (C2×Q8)⋊8(C2×C10), C2.6(C10×C4○D4), (C5×C4⋊C4)⋊66C22, (C2×C4).135(C5×D4), C10.225(C2×C4○D4), (C2×C10).682(C2×D4), C22.30(C5×C4○D4), (C10×C22⋊C4).31C2, (C2×C22⋊C4).11C10, C22⋊C4.10(C2×C10), (C22×C4).53(C2×C10), (C2×C4).12(C22×C10), (C2×C10).230(C4○D4), (C5×C22⋊C4).144C22, SmallGroup(320,1525)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 450 in 322 conjugacy classes, 194 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C5, C2×C4 [×16], C2×C4 [×18], Q8 [×8], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24, C20 [×4], C20 [×10], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×8], C23×C4, C22×Q8, C2×C20 [×16], C2×C20 [×18], C5×Q8 [×8], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22⋊Q8, C5×C22⋊C4 [×8], C5×C4⋊C4 [×12], C22×C20 [×2], C22×C20 [×8], C22×C20 [×4], Q8×C10 [×4], Q8×C10 [×4], C23×C10, C10×C22⋊C4 [×2], C10×C4⋊C4, C10×C4⋊C4 [×2], C5×C22⋊Q8 [×8], C23×C20, Q8×C2×C10, C10×C22⋊Q8
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], Q8 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C2×C10 [×35], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C5×D4 [×4], C5×Q8 [×4], C22×C10 [×15], C2×C22⋊Q8, D4×C10 [×6], Q8×C10 [×6], C5×C4○D4 [×2], C23×C10, C5×C22⋊Q8 [×4], D4×C2×C10, Q8×C2×C10, C10×C4○D4, C10×C22⋊Q8
Generators and relations
G = < a,b,c,d,e | a10=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
►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)
(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 21)(20 22)(31 160)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 157)(39 158)(40 159)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 121)(118 122)(119 123)(120 124)(131 143)(132 144)(133 145)(134 146)(135 147)(136 148)(137 149)(138 150)(139 141)(140 142)
(1 48)(2 49)(3 50)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 21)(20 22)(31 160)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 157)(39 158)(40 159)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 61)(60 62)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 101)(100 102)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 121)(118 122)(119 123)(120 124)(131 143)(132 144)(133 145)(134 146)(135 147)(136 148)(137 149)(138 150)(139 141)(140 142)
(1 79 67 100)(2 80 68 91)(3 71 69 92)(4 72 70 93)(5 73 61 94)(6 74 62 95)(7 75 63 96)(8 76 64 97)(9 77 65 98)(10 78 66 99)(11 148 160 129)(12 149 151 130)(13 150 152 121)(14 141 153 122)(15 142 154 123)(16 143 155 124)(17 144 156 125)(18 145 157 126)(19 146 158 127)(20 147 159 128)(21 134 39 113)(22 135 40 114)(23 136 31 115)(24 137 32 116)(25 138 33 117)(26 139 34 118)(27 140 35 119)(28 131 36 120)(29 132 37 111)(30 133 38 112)(41 86 58 105)(42 87 59 106)(43 88 60 107)(44 89 51 108)(45 90 52 109)(46 81 53 110)(47 82 54 101)(48 83 55 102)(49 84 56 103)(50 85 57 104)
(1 119 67 140)(2 120 68 131)(3 111 69 132)(4 112 70 133)(5 113 61 134)(6 114 62 135)(7 115 63 136)(8 116 64 137)(9 117 65 138)(10 118 66 139)(11 89 160 108)(12 90 151 109)(13 81 152 110)(14 82 153 101)(15 83 154 102)(16 84 155 103)(17 85 156 104)(18 86 157 105)(19 87 158 106)(20 88 159 107)(21 73 39 94)(22 74 40 95)(23 75 31 96)(24 76 32 97)(25 77 33 98)(26 78 34 99)(27 79 35 100)(28 80 36 91)(29 71 37 92)(30 72 38 93)(41 126 58 145)(42 127 59 146)(43 128 60 147)(44 129 51 148)(45 130 52 149)(46 121 53 150)(47 122 54 141)(48 123 55 142)(49 124 56 143)(50 125 57 144)
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), (11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,21)(20,22)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,143)(132,144)(133,145)(134,146)(135,147)(136,148)(137,149)(138,150)(139,141)(140,142), (1,48)(2,49)(3,50)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,21)(20,22)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,61)(60,62)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,101)(100,102)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,143)(132,144)(133,145)(134,146)(135,147)(136,148)(137,149)(138,150)(139,141)(140,142), (1,79,67,100)(2,80,68,91)(3,71,69,92)(4,72,70,93)(5,73,61,94)(6,74,62,95)(7,75,63,96)(8,76,64,97)(9,77,65,98)(10,78,66,99)(11,148,160,129)(12,149,151,130)(13,150,152,121)(14,141,153,122)(15,142,154,123)(16,143,155,124)(17,144,156,125)(18,145,157,126)(19,146,158,127)(20,147,159,128)(21,134,39,113)(22,135,40,114)(23,136,31,115)(24,137,32,116)(25,138,33,117)(26,139,34,118)(27,140,35,119)(28,131,36,120)(29,132,37,111)(30,133,38,112)(41,86,58,105)(42,87,59,106)(43,88,60,107)(44,89,51,108)(45,90,52,109)(46,81,53,110)(47,82,54,101)(48,83,55,102)(49,84,56,103)(50,85,57,104), (1,119,67,140)(2,120,68,131)(3,111,69,132)(4,112,70,133)(5,113,61,134)(6,114,62,135)(7,115,63,136)(8,116,64,137)(9,117,65,138)(10,118,66,139)(11,89,160,108)(12,90,151,109)(13,81,152,110)(14,82,153,101)(15,83,154,102)(16,84,155,103)(17,85,156,104)(18,86,157,105)(19,87,158,106)(20,88,159,107)(21,73,39,94)(22,74,40,95)(23,75,31,96)(24,76,32,97)(25,77,33,98)(26,78,34,99)(27,79,35,100)(28,80,36,91)(29,71,37,92)(30,72,38,93)(41,126,58,145)(42,127,59,146)(43,128,60,147)(44,129,51,148)(45,130,52,149)(46,121,53,150)(47,122,54,141)(48,123,55,142)(49,124,56,143)(50,125,57,144)>;
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), (11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,21)(20,22)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,143)(132,144)(133,145)(134,146)(135,147)(136,148)(137,149)(138,150)(139,141)(140,142), (1,48)(2,49)(3,50)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,21)(20,22)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,61)(60,62)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,101)(100,102)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,143)(132,144)(133,145)(134,146)(135,147)(136,148)(137,149)(138,150)(139,141)(140,142), (1,79,67,100)(2,80,68,91)(3,71,69,92)(4,72,70,93)(5,73,61,94)(6,74,62,95)(7,75,63,96)(8,76,64,97)(9,77,65,98)(10,78,66,99)(11,148,160,129)(12,149,151,130)(13,150,152,121)(14,141,153,122)(15,142,154,123)(16,143,155,124)(17,144,156,125)(18,145,157,126)(19,146,158,127)(20,147,159,128)(21,134,39,113)(22,135,40,114)(23,136,31,115)(24,137,32,116)(25,138,33,117)(26,139,34,118)(27,140,35,119)(28,131,36,120)(29,132,37,111)(30,133,38,112)(41,86,58,105)(42,87,59,106)(43,88,60,107)(44,89,51,108)(45,90,52,109)(46,81,53,110)(47,82,54,101)(48,83,55,102)(49,84,56,103)(50,85,57,104), (1,119,67,140)(2,120,68,131)(3,111,69,132)(4,112,70,133)(5,113,61,134)(6,114,62,135)(7,115,63,136)(8,116,64,137)(9,117,65,138)(10,118,66,139)(11,89,160,108)(12,90,151,109)(13,81,152,110)(14,82,153,101)(15,83,154,102)(16,84,155,103)(17,85,156,104)(18,86,157,105)(19,87,158,106)(20,88,159,107)(21,73,39,94)(22,74,40,95)(23,75,31,96)(24,76,32,97)(25,77,33,98)(26,78,34,99)(27,79,35,100)(28,80,36,91)(29,71,37,92)(30,72,38,93)(41,126,58,145)(42,127,59,146)(43,128,60,147)(44,129,51,148)(45,130,52,149)(46,121,53,150)(47,122,54,141)(48,123,55,142)(49,124,56,143)(50,125,57,144) );
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)], [(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,21),(20,22),(31,160),(32,151),(33,152),(34,153),(35,154),(36,155),(37,156),(38,157),(39,158),(40,159),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,121),(118,122),(119,123),(120,124),(131,143),(132,144),(133,145),(134,146),(135,147),(136,148),(137,149),(138,150),(139,141),(140,142)], [(1,48),(2,49),(3,50),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,21),(20,22),(31,160),(32,151),(33,152),(34,153),(35,154),(36,155),(37,156),(38,157),(39,158),(40,159),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,61),(60,62),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,81),(78,82),(79,83),(80,84),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110),(99,101),(100,102),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,121),(118,122),(119,123),(120,124),(131,143),(132,144),(133,145),(134,146),(135,147),(136,148),(137,149),(138,150),(139,141),(140,142)], [(1,79,67,100),(2,80,68,91),(3,71,69,92),(4,72,70,93),(5,73,61,94),(6,74,62,95),(7,75,63,96),(8,76,64,97),(9,77,65,98),(10,78,66,99),(11,148,160,129),(12,149,151,130),(13,150,152,121),(14,141,153,122),(15,142,154,123),(16,143,155,124),(17,144,156,125),(18,145,157,126),(19,146,158,127),(20,147,159,128),(21,134,39,113),(22,135,40,114),(23,136,31,115),(24,137,32,116),(25,138,33,117),(26,139,34,118),(27,140,35,119),(28,131,36,120),(29,132,37,111),(30,133,38,112),(41,86,58,105),(42,87,59,106),(43,88,60,107),(44,89,51,108),(45,90,52,109),(46,81,53,110),(47,82,54,101),(48,83,55,102),(49,84,56,103),(50,85,57,104)], [(1,119,67,140),(2,120,68,131),(3,111,69,132),(4,112,70,133),(5,113,61,134),(6,114,62,135),(7,115,63,136),(8,116,64,137),(9,117,65,138),(10,118,66,139),(11,89,160,108),(12,90,151,109),(13,81,152,110),(14,82,153,101),(15,83,154,102),(16,84,155,103),(17,85,156,104),(18,86,157,105),(19,87,158,106),(20,88,159,107),(21,73,39,94),(22,74,40,95),(23,75,31,96),(24,76,32,97),(25,77,33,98),(26,78,34,99),(27,79,35,100),(28,80,36,91),(29,71,37,92),(30,72,38,93),(41,126,58,145),(42,127,59,146),(43,128,60,147),(44,129,51,148),(45,130,52,149),(46,121,53,150),(47,122,54,141),(48,123,55,142),(49,124,56,143),(50,125,57,144)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 0 | 25 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 34 | 14 |
| 0 | 0 | 0 | 14 | 7 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,25,0,0,0,0,0,25],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,1,0,0,0,39,1],[1,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,34,14,0,0,0,14,7] >;
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF | 20AG | ··· | 20BL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | Q8 | C4○D4 | C5×D4 | C5×Q8 | C5×C4○D4 |
| kernel | C10×C22⋊Q8 | C10×C22⋊C4 | C10×C4⋊C4 | C5×C22⋊Q8 | C23×C20 | Q8×C2×C10 | C2×C22⋊Q8 | C2×C22⋊C4 | C2×C4⋊C4 | C22⋊Q8 | C23×C4 | C22×Q8 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 3 | 8 | 1 | 1 | 4 | 8 | 12 | 32 | 4 | 4 | 4 | 4 | 4 | 16 | 16 | 16 |
In GAP, Magma, Sage, TeX
C_{10}\times C_2^2\rtimes Q_8 % in TeX
G:=Group("C10xC2^2:Q8"); // GroupNames label
G:=SmallGroup(320,1525);
// by ID
G=gap.SmallGroup(320,1525);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,568,3446]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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