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G = D20.35C23  order 320 = 26·5

16th non-split extension by D20 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.40C24, D20.35C23, 2- (1+4)⋊4D5, Dic10.35C23, C5⋊5(Q8○D8), (C5×D4).39D4, (C5×Q8).39D4, C4○D4.17D10, C20.272(C2×D4), D4⋊D5.2C22, (C2×Q8).93D10, C4.40(C23×D5), Q8⋊D5.3C22, D4.21(C5⋊D4), Q8.Dic5⋊13C2, C5⋊2C8.19C23, Q8.21(C5⋊D4), D4.28(C22×D5), (C5×D4).28C23, D4.D5.3C22, D4.9D10⋊12C2, D4.8D10⋊11C2, Q8.28(C22×D5), (C5×Q8).28C23, C20.C23⋊12C2, C5⋊Q16.4C22, (C2×C20).121C23, C4○D20.34C22, C10.174(C22×D4), (C5×2- (1+4))⋊3C2, D4.10D10⋊10C2, (Q8×C10).154C22, C4.Dic5.32C22, (C2×Dic10).211C22, C4.78(C2×C5⋊D4), (C2×C5⋊Q16)⋊32C2, (C2×C10).88(C2×D4), C22.9(C2×C5⋊D4), C2.47(C22×C5⋊D4), (C5×C4○D4).30C22, (C2×C4).105(C22×D5), (C2×C5⋊2C8).185C22, SmallGroup(320,1510)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — D20.35C23
Chief series
C1 — C5 — C10 — C20 — D20 — C4○D20 — D4.10D10 — D20.35C23
Lower central
C5 — C10 — C20 — D20.35C23

Subgroups: 726 in 248 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×3], C2×C4 [×12], D4, D4 [×3], D4 [×7], Q8, Q8 [×3], Q8 [×9], D5, C10, C10 [×4], C2×C8 [×3], M4(2) [×3], D8, SD16 [×6], Q16 [×9], C2×Q8 [×3], C2×Q8 [×5], C4○D4, C4○D4 [×3], C4○D4 [×9], Dic5 [×3], C20, C20 [×3], C20 [×3], D10, C2×C10 [×3], C2×C10, C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22 [×6], 2- (1+4), 2- (1+4), C5⋊2C8, C5⋊2C8 [×3], Dic10 [×3], Dic10 [×3], C4×D5 [×3], D20, C2×Dic5 [×3], C5⋊D4 [×3], C2×C20 [×3], C2×C20 [×6], C5×D4, C5×D4 [×3], C5×D4 [×3], C5×Q8, C5×Q8 [×3], C5×Q8 [×3], Q8○D8, C2×C5⋊2C8 [×3], C4.Dic5 [×3], D4⋊D5, D4.D5 [×3], Q8⋊D5 [×3], C5⋊Q16 [×9], C2×Dic10 [×3], C4○D20 [×3], D4⋊2D5 [×3], Q8×D5, Q8×C10 [×3], Q8×C10, C5×C4○D4, C5×C4○D4 [×3], C5×C4○D4 [×3], C20.C23 [×3], C2×C5⋊Q16 [×3], Q8.Dic5, D4.8D10 [×3], D4.9D10 [×3], D4.10D10, C5×2- (1+4), D20.35C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C5⋊D4 [×4], C22×D5 [×7], Q8○D8, C2×C5⋊D4 [×6], C23×D5, C22×C5⋊D4, D20.35C23

Generators and relations
 G = < a,b,c,d,e | a20=b2=e2=1, c2=d2=a10, bab=a-1, ac=ca, ad=da, eae=a11, bc=cb, bd=db, ebe=a15b, dcd-1=a10c, ce=ec, de=ed >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

3410000
4000000
0012100
00374000
001033118
00292940
,
1340000
0400000
0012100
0004000
00033118
0000040
,
100000
010000
001412130
0004017
0029372719
00040037
,
4000000
0400000
0030290
0001037
0032251110
00021040
,
100000
010000
0024600
00341700
003852411
0020242617

G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,1,37,10,29,0,0,21,40,33,2,0,0,0,0,1,9,0,0,0,0,18,40],[1,0,0,0,0,0,34,40,0,0,0,0,0,0,1,0,0,0,0,0,21,40,33,0,0,0,0,0,1,0,0,0,0,0,18,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,0,29,0,0,0,12,4,37,40,0,0,13,0,27,0,0,0,0,17,19,37],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,0,32,0,0,0,2,1,25,21,0,0,9,0,11,0,0,0,0,37,10,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,34,38,20,0,0,6,17,5,24,0,0,0,0,24,26,0,0,0,0,11,17] >;
 

56 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E10A10B10C···10L20A···20T
order122222244444444445588888101010···1020···20
size112224202222444202020221010202020224···44···4

56 irreducible representations

dim11111111222222248
type+++++++++++++--
imageC1C2C2C2C2C2C2C2D4D4D5D10D10C5⋊D4C5⋊D4Q8○D8D20.35C23
kernelD20.35C23C20.C23C2×C5⋊Q16Q8.Dic5D4.8D10D4.9D10D4.10D10C5×2- (1+4)C5×D4C5×Q82- (1+4)C2×Q8C4○D4D4Q8C5C1
# reps133133113126812422

In GAP, Magma, Sage, TeX 

D_{20}._{35}C_2^3
 
% in TeX
 
G:=Group("D20.35C2^3");
 
// GroupNames label
 
G:=SmallGroup(320,1510);
 
// by ID
 
G=gap.SmallGroup(320,1510);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,1684,235,102,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^20=b^2=e^2=1,c^2=d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^11,b*c=c*b,b*d=d*b,e*b*e=a^15*b,d*c*d^-1=a^10*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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