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G = C24.65D10  order 320 = 26·5

5th non-split extension by C24 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.65D10, C23.28D20, (C2×C20)⋊35D4, (C23×C4)⋊1D5, (C23×C20)⋊1C2, C10.115(C4×D4), C23.38(C4×D5), C10.70C22≀C2, C2.5(C20⋊7D4), C22.61(C2×D20), C10.80(C4⋊D4), (C22×C10).193D4, (C22×C4).408D10, C2.2(C24⋊2D5), C23.83(C5⋊D4), C5⋊5(C23.23D4), C22⋊2(D10⋊C4), C22.64(C4○D20), (C23×D5).24C22, C23.304(C22×D5), C10.10C42⋊25C2, (C22×C20).485C22, (C23×C10).100C22, (C22×C10).364C23, C10.69(C22.D4), C2.5(C23.23D10), (C22×Dic5).67C22, (C2×C5⋊D4)⋊12C4, C2.29(C4×C5⋊D4), (C2×C4)⋊15(C5⋊D4), (C22×D5)⋊7(C2×C4), (C2×C23.D5)⋊7C2, C22.150(C2×C4×D5), (C2×C10)⋊8(C22⋊C4), (C2×Dic5)⋊11(C2×C4), (C2×C10).550(C2×D4), (C2×D10⋊C4)⋊11C2, (C22×C5⋊D4).7C2, C22.88(C2×C5⋊D4), (C2×C10).92(C4○D4), C2.36(C2×D10⋊C4), C10.106(C2×C22⋊C4), (C2×C10).244(C22×C4), (C22×C10).168(C2×C4), SmallGroup(320,840)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C24.65D10
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C22×C5⋊D4 — C24.65D10
Lower central
C5 — C2×C10 — C24.65D10
Upper central
C1 — C23 — C23×C4

Generators and relations for C24.65D10
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 1022 in 286 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C23×C4, C22×D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C23.23D4, D10⋊C4, C23.D5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C22×C20, C23×D5, C23×C10, C10.10C42, C2×D10⋊C4, C2×C23.D5, C22×C5⋊D4, C23×C20, C24.65D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, C4○D4, D10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C4×D5, D20, C5⋊D4, C22×D5, C23.23D4, D10⋊C4, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C2×D10⋊C4, C4×C5⋊D4, C23.23D10, C20⋊7D4, C24⋊2D5, C24.65D10

Smallest permutation representation of C24.65D10
►On 160 points
Generators in S160
(1 139)(2 140)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 61)(57 62)(58 63)(59 64)(60 65)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 141)(108 142)(109 143)(110 144)(111 145)(112 146)(113 147)(114 148)(115 149)(116 150)(117 151)(118 152)(119 153)(120 154)

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

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

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

92 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4N5A5B10A···10AD20A···20AF
order12···22222224···44···45510···1020···20
size11···1222220202···220···20222···22···2

92 irreducible representations

dim111111122222222222
type++++++++++++
imageC1C2C2C2C2C2C4D4D4D5C4○D4D10D10C5⋊D4C4×D5D20C5⋊D4C4○D20
kernelC24.65D10C10.10C42C2×D10⋊C4C2×C23.D5C22×C5⋊D4C23×C20C2×C5⋊D4C2×C20C22×C10C23×C4C2×C10C22×C4C24C2×C4C23C23C23C22
# reps12211184424421688816

Matrix representation of C24.65D10 ►in GL6(𝔽41)

2360000
35180000
0023600
00351800
0000181
0000523
,
100000
010000
0040000
0004000
000010
000001
,
4000000
0400000
0040000
0004000
0000400
0000040
,
0400000
1350000
000900
00321300
00002511
00001439
,
0400000
4000000
000900
009000
0000211
00003739

G:=sub<GL(6,GF(41))| [23,35,0,0,0,0,6,18,0,0,0,0,0,0,23,35,0,0,0,0,6,18,0,0,0,0,0,0,18,5,0,0,0,0,1,23],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,35,0,0,0,0,0,0,0,32,0,0,0,0,9,13,0,0,0,0,0,0,25,14,0,0,0,0,11,39],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,2,37,0,0,0,0,11,39] >;
 

C24.65D10 in GAP, Magma, Sage, TeX 

C_2^4._{65}D_{10}
 
% in TeX
 
G:=Group("C2^4.65D10");
 
// GroupNames label
 
G:=SmallGroup(320,840);
 
// by ID
 
G=gap.SmallGroup(320,840);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,58,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
 
// generators/relations
 

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