G = C10.422- 1+4 order 320 = 26·5
42nd non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.422- 1+4, (Q8×C10)⋊22C4, (C2×Q8)⋊9Dic5, (Q8×Dic5)⋊25C2, Q8.8(C2×Dic5), (C22×Q8).9D5, C10.68(C23×C4), (C2×Q8).207D10, C2.9(C23×Dic5), (C2×C10).304C24, C20.155(C22×C4), (C2×C20).551C23, (C22×C4).276D10, C4.19(C22×Dic5), C22.47(C23×D5), C4⋊Dic5.389C22, (Q8×C10).233C22, C23.237(C22×D5), (C22×C20).284C22, (C22×C10).422C23, C2.4(Q8.10D10), C5⋊4(C23.32C23), (C2×Dic5).298C23, (C4×Dic5).177C22, C23.D5.145C22, C22.10(C22×Dic5), C23.21D10.25C2, (Q8×C2×C10).9C2, (C5×Q8).41(C2×C4), (C2×C20).307(C2×C4), (C2×C4).30(C2×Dic5), (C2×C4).632(C22×D5), (C2×C10).311(C22×C4), SmallGroup(320,1484)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.422- 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=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 558 in 266 conjugacy classes, 191 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C2×C10, C42⋊C2, C4×Q8, C22×Q8, C2×Dic5, C2×C20, C5×Q8, C22×C10, C23.32C23, C4×Dic5, C4⋊Dic5, C23.D5, C22×C20, Q8×C10, C23.21D10, Q8×Dic5, Q8×C2×C10, C10.422- 1+4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, Dic5, D10, C23×C4, 2- 1+4, C2×Dic5, C22×D5, C23.32C23, C22×Dic5, C23×D5, Q8.10D10, C23×Dic5, C10.422- 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 65 30 53)(2 66 21 54)(3 67 22 55)(4 68 23 56)(5 69 24 57)(6 70 25 58)(7 61 26 59)(8 62 27 60)(9 63 28 51)(10 64 29 52)(11 118 153 121)(12 119 154 122)(13 120 155 123)(14 111 156 124)(15 112 157 125)(16 113 158 126)(17 114 159 127)(18 115 160 128)(19 116 151 129)(20 117 152 130)(31 89 43 76)(32 90 44 77)(33 81 45 78)(34 82 46 79)(35 83 47 80)(36 84 48 71)(37 85 49 72)(38 86 50 73)(39 87 41 74)(40 88 42 75)(91 136 104 149)(92 137 105 150)(93 138 106 141)(94 139 107 142)(95 140 108 143)(96 131 109 144)(97 132 110 145)(98 133 101 146)(99 134 102 147)(100 135 103 148)
(1 150 6 145)(2 149 7 144)(3 148 8 143)(4 147 9 142)(5 146 10 141)(11 37 16 32)(12 36 17 31)(13 35 18 40)(14 34 19 39)(15 33 20 38)(21 136 26 131)(22 135 27 140)(23 134 28 139)(24 133 29 138)(25 132 30 137)(41 156 46 151)(42 155 47 160)(43 154 48 159)(44 153 49 158)(45 152 50 157)(51 94 56 99)(52 93 57 98)(53 92 58 97)(54 91 59 96)(55 100 60 95)(61 109 66 104)(62 108 67 103)(63 107 68 102)(64 106 69 101)(65 105 70 110)(71 114 76 119)(72 113 77 118)(73 112 78 117)(74 111 79 116)(75 120 80 115)(81 130 86 125)(82 129 87 124)(83 128 88 123)(84 127 89 122)(85 126 90 121)
(1 117 25 125)(2 116 26 124)(3 115 27 123)(4 114 28 122)(5 113 29 121)(6 112 30 130)(7 111 21 129)(8 120 22 128)(9 119 23 127)(10 118 24 126)(11 69 158 52)(12 68 159 51)(13 67 160 60)(14 66 151 59)(15 65 152 58)(16 64 153 57)(17 63 154 56)(18 62 155 55)(19 61 156 54)(20 70 157 53)(31 107 48 99)(32 106 49 98)(33 105 50 97)(34 104 41 96)(35 103 42 95)(36 102 43 94)(37 101 44 93)(38 110 45 92)(39 109 46 91)(40 108 47 100)(71 134 89 142)(72 133 90 141)(73 132 81 150)(74 131 82 149)(75 140 83 148)(76 139 84 147)(77 138 85 146)(78 137 86 145)(79 136 87 144)(80 135 88 143)
(1 70 30 58)(2 61 21 59)(3 62 22 60)(4 63 23 51)(5 64 24 52)(6 65 25 53)(7 66 26 54)(8 67 27 55)(9 68 28 56)(10 69 29 57)(11 126 153 113)(12 127 154 114)(13 128 155 115)(14 129 156 116)(15 130 157 117)(16 121 158 118)(17 122 159 119)(18 123 160 120)(19 124 151 111)(20 125 152 112)(31 84 43 71)(32 85 44 72)(33 86 45 73)(34 87 46 74)(35 88 47 75)(36 89 48 76)(37 90 49 77)(38 81 50 78)(39 82 41 79)(40 83 42 80)(91 144 104 131)(92 145 105 132)(93 146 106 133)(94 147 107 134)(95 148 108 135)(96 149 109 136)(97 150 110 137)(98 141 101 138)(99 142 102 139)(100 143 103 140)
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), (1,65,30,53)(2,66,21,54)(3,67,22,55)(4,68,23,56)(5,69,24,57)(6,70,25,58)(7,61,26,59)(8,62,27,60)(9,63,28,51)(10,64,29,52)(11,118,153,121)(12,119,154,122)(13,120,155,123)(14,111,156,124)(15,112,157,125)(16,113,158,126)(17,114,159,127)(18,115,160,128)(19,116,151,129)(20,117,152,130)(31,89,43,76)(32,90,44,77)(33,81,45,78)(34,82,46,79)(35,83,47,80)(36,84,48,71)(37,85,49,72)(38,86,50,73)(39,87,41,74)(40,88,42,75)(91,136,104,149)(92,137,105,150)(93,138,106,141)(94,139,107,142)(95,140,108,143)(96,131,109,144)(97,132,110,145)(98,133,101,146)(99,134,102,147)(100,135,103,148), (1,150,6,145)(2,149,7,144)(3,148,8,143)(4,147,9,142)(5,146,10,141)(11,37,16,32)(12,36,17,31)(13,35,18,40)(14,34,19,39)(15,33,20,38)(21,136,26,131)(22,135,27,140)(23,134,28,139)(24,133,29,138)(25,132,30,137)(41,156,46,151)(42,155,47,160)(43,154,48,159)(44,153,49,158)(45,152,50,157)(51,94,56,99)(52,93,57,98)(53,92,58,97)(54,91,59,96)(55,100,60,95)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,114,76,119)(72,113,77,118)(73,112,78,117)(74,111,79,116)(75,120,80,115)(81,130,86,125)(82,129,87,124)(83,128,88,123)(84,127,89,122)(85,126,90,121), (1,117,25,125)(2,116,26,124)(3,115,27,123)(4,114,28,122)(5,113,29,121)(6,112,30,130)(7,111,21,129)(8,120,22,128)(9,119,23,127)(10,118,24,126)(11,69,158,52)(12,68,159,51)(13,67,160,60)(14,66,151,59)(15,65,152,58)(16,64,153,57)(17,63,154,56)(18,62,155,55)(19,61,156,54)(20,70,157,53)(31,107,48,99)(32,106,49,98)(33,105,50,97)(34,104,41,96)(35,103,42,95)(36,102,43,94)(37,101,44,93)(38,110,45,92)(39,109,46,91)(40,108,47,100)(71,134,89,142)(72,133,90,141)(73,132,81,150)(74,131,82,149)(75,140,83,148)(76,139,84,147)(77,138,85,146)(78,137,86,145)(79,136,87,144)(80,135,88,143), (1,70,30,58)(2,61,21,59)(3,62,22,60)(4,63,23,51)(5,64,24,52)(6,65,25,53)(7,66,26,54)(8,67,27,55)(9,68,28,56)(10,69,29,57)(11,126,153,113)(12,127,154,114)(13,128,155,115)(14,129,156,116)(15,130,157,117)(16,121,158,118)(17,122,159,119)(18,123,160,120)(19,124,151,111)(20,125,152,112)(31,84,43,71)(32,85,44,72)(33,86,45,73)(34,87,46,74)(35,88,47,75)(36,89,48,76)(37,90,49,77)(38,81,50,78)(39,82,41,79)(40,83,42,80)(91,144,104,131)(92,145,105,132)(93,146,106,133)(94,147,107,134)(95,148,108,135)(96,149,109,136)(97,150,110,137)(98,141,101,138)(99,142,102,139)(100,143,103,140)>;
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), (1,65,30,53)(2,66,21,54)(3,67,22,55)(4,68,23,56)(5,69,24,57)(6,70,25,58)(7,61,26,59)(8,62,27,60)(9,63,28,51)(10,64,29,52)(11,118,153,121)(12,119,154,122)(13,120,155,123)(14,111,156,124)(15,112,157,125)(16,113,158,126)(17,114,159,127)(18,115,160,128)(19,116,151,129)(20,117,152,130)(31,89,43,76)(32,90,44,77)(33,81,45,78)(34,82,46,79)(35,83,47,80)(36,84,48,71)(37,85,49,72)(38,86,50,73)(39,87,41,74)(40,88,42,75)(91,136,104,149)(92,137,105,150)(93,138,106,141)(94,139,107,142)(95,140,108,143)(96,131,109,144)(97,132,110,145)(98,133,101,146)(99,134,102,147)(100,135,103,148), (1,150,6,145)(2,149,7,144)(3,148,8,143)(4,147,9,142)(5,146,10,141)(11,37,16,32)(12,36,17,31)(13,35,18,40)(14,34,19,39)(15,33,20,38)(21,136,26,131)(22,135,27,140)(23,134,28,139)(24,133,29,138)(25,132,30,137)(41,156,46,151)(42,155,47,160)(43,154,48,159)(44,153,49,158)(45,152,50,157)(51,94,56,99)(52,93,57,98)(53,92,58,97)(54,91,59,96)(55,100,60,95)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,114,76,119)(72,113,77,118)(73,112,78,117)(74,111,79,116)(75,120,80,115)(81,130,86,125)(82,129,87,124)(83,128,88,123)(84,127,89,122)(85,126,90,121), (1,117,25,125)(2,116,26,124)(3,115,27,123)(4,114,28,122)(5,113,29,121)(6,112,30,130)(7,111,21,129)(8,120,22,128)(9,119,23,127)(10,118,24,126)(11,69,158,52)(12,68,159,51)(13,67,160,60)(14,66,151,59)(15,65,152,58)(16,64,153,57)(17,63,154,56)(18,62,155,55)(19,61,156,54)(20,70,157,53)(31,107,48,99)(32,106,49,98)(33,105,50,97)(34,104,41,96)(35,103,42,95)(36,102,43,94)(37,101,44,93)(38,110,45,92)(39,109,46,91)(40,108,47,100)(71,134,89,142)(72,133,90,141)(73,132,81,150)(74,131,82,149)(75,140,83,148)(76,139,84,147)(77,138,85,146)(78,137,86,145)(79,136,87,144)(80,135,88,143), (1,70,30,58)(2,61,21,59)(3,62,22,60)(4,63,23,51)(5,64,24,52)(6,65,25,53)(7,66,26,54)(8,67,27,55)(9,68,28,56)(10,69,29,57)(11,126,153,113)(12,127,154,114)(13,128,155,115)(14,129,156,116)(15,130,157,117)(16,121,158,118)(17,122,159,119)(18,123,160,120)(19,124,151,111)(20,125,152,112)(31,84,43,71)(32,85,44,72)(33,86,45,73)(34,87,46,74)(35,88,47,75)(36,89,48,76)(37,90,49,77)(38,81,50,78)(39,82,41,79)(40,83,42,80)(91,144,104,131)(92,145,105,132)(93,146,106,133)(94,147,107,134)(95,148,108,135)(96,149,109,136)(97,150,110,137)(98,141,101,138)(99,142,102,139)(100,143,103,140) );
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)], [(1,65,30,53),(2,66,21,54),(3,67,22,55),(4,68,23,56),(5,69,24,57),(6,70,25,58),(7,61,26,59),(8,62,27,60),(9,63,28,51),(10,64,29,52),(11,118,153,121),(12,119,154,122),(13,120,155,123),(14,111,156,124),(15,112,157,125),(16,113,158,126),(17,114,159,127),(18,115,160,128),(19,116,151,129),(20,117,152,130),(31,89,43,76),(32,90,44,77),(33,81,45,78),(34,82,46,79),(35,83,47,80),(36,84,48,71),(37,85,49,72),(38,86,50,73),(39,87,41,74),(40,88,42,75),(91,136,104,149),(92,137,105,150),(93,138,106,141),(94,139,107,142),(95,140,108,143),(96,131,109,144),(97,132,110,145),(98,133,101,146),(99,134,102,147),(100,135,103,148)], [(1,150,6,145),(2,149,7,144),(3,148,8,143),(4,147,9,142),(5,146,10,141),(11,37,16,32),(12,36,17,31),(13,35,18,40),(14,34,19,39),(15,33,20,38),(21,136,26,131),(22,135,27,140),(23,134,28,139),(24,133,29,138),(25,132,30,137),(41,156,46,151),(42,155,47,160),(43,154,48,159),(44,153,49,158),(45,152,50,157),(51,94,56,99),(52,93,57,98),(53,92,58,97),(54,91,59,96),(55,100,60,95),(61,109,66,104),(62,108,67,103),(63,107,68,102),(64,106,69,101),(65,105,70,110),(71,114,76,119),(72,113,77,118),(73,112,78,117),(74,111,79,116),(75,120,80,115),(81,130,86,125),(82,129,87,124),(83,128,88,123),(84,127,89,122),(85,126,90,121)], [(1,117,25,125),(2,116,26,124),(3,115,27,123),(4,114,28,122),(5,113,29,121),(6,112,30,130),(7,111,21,129),(8,120,22,128),(9,119,23,127),(10,118,24,126),(11,69,158,52),(12,68,159,51),(13,67,160,60),(14,66,151,59),(15,65,152,58),(16,64,153,57),(17,63,154,56),(18,62,155,55),(19,61,156,54),(20,70,157,53),(31,107,48,99),(32,106,49,98),(33,105,50,97),(34,104,41,96),(35,103,42,95),(36,102,43,94),(37,101,44,93),(38,110,45,92),(39,109,46,91),(40,108,47,100),(71,134,89,142),(72,133,90,141),(73,132,81,150),(74,131,82,149),(75,140,83,148),(76,139,84,147),(77,138,85,146),(78,137,86,145),(79,136,87,144),(80,135,88,143)], [(1,70,30,58),(2,61,21,59),(3,62,22,60),(4,63,23,51),(5,64,24,52),(6,65,25,53),(7,66,26,54),(8,67,27,55),(9,68,28,56),(10,69,29,57),(11,126,153,113),(12,127,154,114),(13,128,155,115),(14,129,156,116),(15,130,157,117),(16,121,158,118),(17,122,159,119),(18,123,160,120),(19,124,151,111),(20,125,152,112),(31,84,43,71),(32,85,44,72),(33,86,45,73),(34,87,46,74),(35,88,47,75),(36,89,48,76),(37,90,49,77),(38,81,50,78),(39,82,41,79),(40,83,42,80),(91,144,104,131),(92,145,105,132),(93,146,106,133),(94,147,107,134),(95,148,108,135),(96,149,109,136),(97,150,110,137),(98,141,101,138),(99,142,102,139),(100,143,103,140)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4AB | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | D5 | D10 | Dic5 | D10 | 2- 1+4 | Q8.10D10 |
| kernel | C10.422- 1+4 | C23.21D10 | Q8×Dic5 | Q8×C2×C10 | Q8×C10 | C22×Q8 | C22×C4 | C2×Q8 | C2×Q8 | C10 | C2 |
| # reps | 1 | 6 | 8 | 1 | 16 | 2 | 6 | 16 | 8 | 2 | 8 |
Matrix representation of C10.422- 1+4 ►in GL6(𝔽41)
| 23 | 0 | 0 | 0 | 0 | 0 |
| 17 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 3 | 40 | 0 | 18 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 3 | 20 | 0 | 0 |
| 0 | 0 | 9 | 38 | 23 | 6 |
| 0 | 0 | 38 | 17 | 21 | 18 |
| 36 | 38 | 0 | 0 | 0 | 0 |
| 36 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 38 | 23 | 6 |
| 0 | 0 | 19 | 21 | 17 | 40 |
| 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 25 | 24 | 15 | 11 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 5 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 40 | 1 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 36 | 2 | 40 | 38 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 3 | 20 | 0 | 0 |
| 0 | 0 | 32 | 3 | 18 | 35 |
| 0 | 0 | 25 | 35 | 20 | 23 |
G:=sub<GL(6,GF(41))| [23,17,0,0,0,0,0,25,0,0,0,0,0,0,16,0,0,3,0,0,0,16,0,40,0,0,0,0,18,0,0,0,0,0,0,18],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,21,3,9,38,0,0,3,20,38,17,0,0,0,0,23,21,0,0,0,0,6,18],[36,36,0,0,0,0,38,5,0,0,0,0,0,0,9,19,21,25,0,0,38,21,3,24,0,0,23,17,0,15,0,0,6,40,0,11],[5,5,0,0,0,0,3,36,0,0,0,0,0,0,3,0,0,36,0,0,40,0,40,2,0,0,1,1,0,40,0,0,2,0,0,38],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,3,32,25,0,0,3,20,3,35,0,0,0,0,18,20,0,0,0,0,35,23] >;
C10.422- 1+4 in GAP, Magma, Sage, TeX
C_{10}._{42}2_-^{1+4} % in TeX
G:=Group("C10.42ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1484);
// by ID
G=gap.SmallGroup(320,1484);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,1123,12550]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations