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G = C10.812- 1+4order 320 = 26·5

36th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.812- 1+4, C10.602+ 1+4, C20⋊Q831C2, C4⋊C4.105D10, (C2×D4).97D10, C22⋊C4.26D10, C20.48D414C2, (C2×C20).178C23, (C2×C10).195C24, (C22×C4).256D10, C2.62(D46D10), C22.D4.3D5, Dic5.Q826C2, C20.17D4.10C2, (D4×C10).133C22, C23.D1029C2, C4⋊Dic5.226C22, (C22×C20).86C22, C22.216(C23×D5), C23.128(C22×D5), Dic5.14D430C2, C23.D5.41C22, (C22×C10).220C23, C52(C22.57C24), (C2×Dic10).38C22, (C2×Dic5).100C23, (C4×Dic5).130C22, C10.D4.40C22, C23.18D10.3C2, C2.42(D4.10D10), (C22×Dic5).128C22, (C2×C4).59(C22×D5), (C5×C4⋊C4).175C22, (C5×C22⋊C4).50C22, (C5×C22.D4).3C2, SmallGroup(320,1323)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.812- 1+4
C1C5C10C2×C10C2×Dic5C22×Dic5Dic5.14D4 — C10.812- 1+4
C5C2×C10 — C10.812- 1+4
C1C22C22.D4

Generators and relations for C10.812- 1+4
 G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=e2=a5b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=b-1, bd=db, ebe-1=a5b, dcd-1=a5c, ce=ec, ede-1=a5b2d >

Subgroups: 614 in 196 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4, C2×C4 [×4], C2×C4 [×10], D4, Q8 [×3], C23 [×2], C10, C10 [×2], C10 [×2], C42 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4, C2×D4, C2×Q8 [×3], Dic5 [×8], C20 [×5], C2×C10, C2×C10 [×6], C22⋊Q8 [×4], C22.D4, C22.D4, C4.4D4, C42.C2 [×2], C422C2 [×4], C4⋊Q8 [×2], Dic10 [×3], C2×Dic5 [×8], C2×Dic5, C2×C20, C2×C20 [×4], C2×C20, C5×D4, C22×C10 [×2], C22.57C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×10], C4⋊Dic5 [×4], C23.D5, C23.D5 [×6], C5×C22⋊C4, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C22×Dic5, C22×C20, D4×C10, Dic5.14D4 [×2], C23.D10 [×4], C20⋊Q8 [×2], Dic5.Q8 [×2], C20.48D4 [×2], C23.18D10, C20.17D4, C5×C22.D4, C10.812- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4, 2- 1+4 [×2], C22×D5 [×7], C22.57C24, C23×D5, D46D10, D4.10D10 [×2], C10.812- 1+4

Smallest permutation representation of C10.812- 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)
(1 150 25 132)(2 149 26 131)(3 148 27 140)(4 147 28 139)(5 146 29 138)(6 145 30 137)(7 144 21 136)(8 143 22 135)(9 142 23 134)(10 141 24 133)(11 37 158 44)(12 36 159 43)(13 35 160 42)(14 34 151 41)(15 33 152 50)(16 32 153 49)(17 31 154 48)(18 40 155 47)(19 39 156 46)(20 38 157 45)(51 99 68 107)(52 98 69 106)(53 97 70 105)(54 96 61 104)(55 95 62 103)(56 94 63 102)(57 93 64 101)(58 92 65 110)(59 91 66 109)(60 100 67 108)(71 122 89 114)(72 121 90 113)(73 130 81 112)(74 129 82 111)(75 128 83 120)(76 127 84 119)(77 126 85 118)(78 125 86 117)(79 124 87 116)(80 123 88 115)
(1 112 6 117)(2 111 7 116)(3 120 8 115)(4 119 9 114)(5 118 10 113)(11 64 16 69)(12 63 17 68)(13 62 18 67)(14 61 19 66)(15 70 20 65)(21 124 26 129)(22 123 27 128)(23 122 28 127)(24 121 29 126)(25 130 30 125)(31 99 36 94)(32 98 37 93)(33 97 38 92)(34 96 39 91)(35 95 40 100)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(51 159 56 154)(52 158 57 153)(53 157 58 152)(54 156 59 151)(55 155 60 160)(71 139 76 134)(72 138 77 133)(73 137 78 132)(74 136 79 131)(75 135 80 140)(81 145 86 150)(82 144 87 149)(83 143 88 148)(84 142 89 147)(85 141 90 146)
(1 45 30 33)(2 46 21 34)(3 47 22 35)(4 48 23 36)(5 49 24 37)(6 50 25 38)(7 41 26 39)(8 42 27 40)(9 43 28 31)(10 44 29 32)(11 138 153 141)(12 139 154 142)(13 140 155 143)(14 131 156 144)(15 132 157 145)(16 133 158 146)(17 134 159 147)(18 135 160 148)(19 136 151 149)(20 137 152 150)(51 84 63 71)(52 85 64 72)(53 86 65 73)(54 87 66 74)(55 88 67 75)(56 89 68 76)(57 90 69 77)(58 81 70 78)(59 82 61 79)(60 83 62 80)(91 111 104 124)(92 112 105 125)(93 113 106 126)(94 114 107 127)(95 115 108 128)(96 116 109 129)(97 117 110 130)(98 118 101 121)(99 119 102 122)(100 120 103 123)
(1 157 30 15)(2 156 21 14)(3 155 22 13)(4 154 23 12)(5 153 24 11)(6 152 25 20)(7 151 26 19)(8 160 27 18)(9 159 28 17)(10 158 29 16)(31 147 43 134)(32 146 44 133)(33 145 45 132)(34 144 46 131)(35 143 47 140)(36 142 48 139)(37 141 49 138)(38 150 50 137)(39 149 41 136)(40 148 42 135)(51 122 63 119)(52 121 64 118)(53 130 65 117)(54 129 66 116)(55 128 67 115)(56 127 68 114)(57 126 69 113)(58 125 70 112)(59 124 61 111)(60 123 62 120)(71 99 84 102)(72 98 85 101)(73 97 86 110)(74 96 87 109)(75 95 88 108)(76 94 89 107)(77 93 90 106)(78 92 81 105)(79 91 82 104)(80 100 83 103)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A···4E4F···4M5A5B10A···10F10G10H10I10J10K10L20A···20H20I···20N
order1222224···44···45510···1010101010101020···2020···20
size1111444···420···20222···24444884···48···8

47 irreducible representations

dim111111111222224444
type+++++++++++++++--
imageC1C2C2C2C2C2C2C2C2D5D10D10D10D102+ 1+42- 1+4D46D10D4.10D10
kernelC10.812- 1+4Dic5.14D4C23.D10C20⋊Q8Dic5.Q8C20.48D4C23.18D10C20.17D4C5×C22.D4C22.D4C22⋊C4C4⋊C4C22×C4C2×D4C10C10C2C2
# reps124222111264221248

Matrix representation of C10.812- 1+4 in GL8(𝔽41)

3535000000
640000000
000350000
007340000
000040000
000004000
000000400
000000040
,
3927000000
152000000
0028390000
003130000
00001501537
00001502626
000015392839
00000132839
,
27393330000
37143950000
312523140000
37359180000
0000151500
0000152600
0000028213
000015282839
,
4003200000
0401200000
401100000
396010000
000040200
00000100
000040101
000040110
,
2142180000
26396120000
0028390000
003130000
00001503715
00001502626
000015283928
0000023928

G:=sub<GL(8,GF(41))| [35,6,0,0,0,0,0,0,35,40,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[39,15,0,0,0,0,0,0,27,2,0,0,0,0,0,0,0,0,28,3,0,0,0,0,0,0,39,13,0,0,0,0,0,0,0,0,15,15,15,0,0,0,0,0,0,0,39,13,0,0,0,0,15,26,28,28,0,0,0,0,37,26,39,39],[27,37,31,37,0,0,0,0,39,14,25,35,0,0,0,0,3,39,23,9,0,0,0,0,33,5,14,18,0,0,0,0,0,0,0,0,15,15,0,15,0,0,0,0,15,26,28,28,0,0,0,0,0,0,2,28,0,0,0,0,0,0,13,39],[40,0,40,39,0,0,0,0,0,40,1,6,0,0,0,0,3,1,1,0,0,0,0,0,20,20,0,1,0,0,0,0,0,0,0,0,40,0,40,40,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[2,26,0,0,0,0,0,0,14,39,0,0,0,0,0,0,21,6,28,3,0,0,0,0,8,12,39,13,0,0,0,0,0,0,0,0,15,15,15,0,0,0,0,0,0,0,28,2,0,0,0,0,37,26,39,39,0,0,0,0,15,26,28,28] >;

C10.812- 1+4 in GAP, Magma, Sage, TeX

C_{10}._{81}2_-^{1+4}
% in TeX

G:=Group("C10.81ES-(2,2)");
// GroupNames label

G:=SmallGroup(320,1323);
// by ID

G=gap.SmallGroup(320,1323);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,219,184,1571,570,12550]);
// Polycyclic

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

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