metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.23D18, (C2×D4).5D9, (C2×C18).7D4, (C6×D4).19S3, (C2×C4).15D18, C18.48(C2×D4), Dic9⋊C4⋊14C2, (D4×C18).10C2, (C2×C12).215D6, (C22×C6).48D6, C18.29(C4○D4), C18.D4⋊8C2, (C2×C18).50C23, (C2×C36).60C22, (C22×Dic9)⋊5C2, C22.4(C9⋊D4), C9⋊5(C22.D4), C6.86(D4⋊2S3), C2.15(D4⋊2D9), C3.(C23.23D6), C22.57(C22×D9), (C22×C18).18C22, (C2×Dic9).15C22, C2.11(C2×C9⋊D4), C6.95(C2×C3⋊D4), (C2×C6).4(C3⋊D4), (C2×C6).207(C22×S3), SmallGroup(288,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.23D18
G = < a,b,c,d,e | a2=b2=c2=d18=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 404 in 117 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C9, Dic3 [×4], C12, C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C18, C18 [×2], C18 [×3], C2×Dic3 [×6], C2×C12, C3×D4 [×2], C22×C6 [×2], C22.D4, Dic9 [×4], C36, C2×C18, C2×C18 [×2], C2×C18 [×5], Dic3⋊C4 [×2], C6.D4 [×3], C22×Dic3, C6×D4, C2×Dic9 [×4], C2×Dic9 [×2], C2×C36, D4×C9 [×2], C22×C18 [×2], C23.23D6, Dic9⋊C4 [×2], C18.D4, C18.D4 [×2], C22×Dic9, D4×C18, C23.23D18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D9, C3⋊D4 [×2], C22×S3, C22.D4, D18 [×3], D4⋊2S3 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C23.23D6, D4⋊2D9 [×2], C2×C9⋊D4, C23.23D18
►On 144 points
Generators in S144
(1 45)(2 37)(3 47)(4 39)(5 49)(6 41)(7 51)(8 43)(9 53)(10 48)(11 40)(12 50)(13 42)(14 52)(15 44)(16 54)(17 46)(18 38)(19 65)(20 57)(21 67)(22 59)(23 69)(24 61)(25 71)(26 63)(27 55)(28 56)(29 66)(30 58)(31 68)(32 60)(33 70)(34 62)(35 72)(36 64)(73 122)(74 139)(75 124)(76 141)(77 126)(78 143)(79 110)(80 127)(81 112)(82 129)(83 114)(84 131)(85 116)(86 133)(87 118)(88 135)(89 120)(90 137)(91 119)(92 136)(93 121)(94 138)(95 123)(96 140)(97 125)(98 142)(99 109)(100 144)(101 111)(102 128)(103 113)(104 130)(105 115)(106 132)(107 117)(108 134)
(1 26)(2 27)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 28)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)(109 134)(110 135)(111 136)(112 137)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)
(1 16)(2 17)(3 18)(4 10)(5 11)(6 12)(7 13)(8 14)(9 15)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 91)(89 92)(90 93)(109 143)(110 144)(111 127)(112 128)(113 129)(114 130)(115 131)(116 132)(117 133)(118 134)(119 135)(120 136)(121 137)(122 138)(123 139)(124 140)(125 141)(126 142)
(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)
(1 103 35 73)(2 93 36 81)(3 101 28 89)(4 91 29 79)(5 99 30 87)(6 107 31 77)(7 97 32 85)(8 105 33 75)(9 95 34 83)(10 88 20 100)(11 78 21 108)(12 86 22 98)(13 76 23 106)(14 84 24 96)(15 74 25 104)(16 82 26 94)(17 90 27 102)(18 80 19 92)(37 137 64 128)(38 111 65 120)(39 135 66 144)(40 109 67 118)(41 133 68 142)(42 125 69 116)(43 131 70 140)(44 123 71 114)(45 129 72 138)(46 121 55 112)(47 127 56 136)(48 119 57 110)(49 143 58 134)(50 117 59 126)(51 141 60 132)(52 115 61 124)(53 139 62 130)(54 113 63 122)
G:=sub<Sym(144)| (1,45)(2,37)(3,47)(4,39)(5,49)(6,41)(7,51)(8,43)(9,53)(10,48)(11,40)(12,50)(13,42)(14,52)(15,44)(16,54)(17,46)(18,38)(19,65)(20,57)(21,67)(22,59)(23,69)(24,61)(25,71)(26,63)(27,55)(28,56)(29,66)(30,58)(31,68)(32,60)(33,70)(34,62)(35,72)(36,64)(73,122)(74,139)(75,124)(76,141)(77,126)(78,143)(79,110)(80,127)(81,112)(82,129)(83,114)(84,131)(85,116)(86,133)(87,118)(88,135)(89,120)(90,137)(91,119)(92,136)(93,121)(94,138)(95,123)(96,140)(97,125)(98,142)(99,109)(100,144)(101,111)(102,128)(103,113)(104,130)(105,115)(106,132)(107,117)(108,134), (1,26)(2,27)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,28)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133), (1,16)(2,17)(3,18)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,91)(89,92)(90,93)(109,143)(110,144)(111,127)(112,128)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136)(121,137)(122,138)(123,139)(124,140)(125,141)(126,142), (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), (1,103,35,73)(2,93,36,81)(3,101,28,89)(4,91,29,79)(5,99,30,87)(6,107,31,77)(7,97,32,85)(8,105,33,75)(9,95,34,83)(10,88,20,100)(11,78,21,108)(12,86,22,98)(13,76,23,106)(14,84,24,96)(15,74,25,104)(16,82,26,94)(17,90,27,102)(18,80,19,92)(37,137,64,128)(38,111,65,120)(39,135,66,144)(40,109,67,118)(41,133,68,142)(42,125,69,116)(43,131,70,140)(44,123,71,114)(45,129,72,138)(46,121,55,112)(47,127,56,136)(48,119,57,110)(49,143,58,134)(50,117,59,126)(51,141,60,132)(52,115,61,124)(53,139,62,130)(54,113,63,122)>;
G:=Group( (1,45)(2,37)(3,47)(4,39)(5,49)(6,41)(7,51)(8,43)(9,53)(10,48)(11,40)(12,50)(13,42)(14,52)(15,44)(16,54)(17,46)(18,38)(19,65)(20,57)(21,67)(22,59)(23,69)(24,61)(25,71)(26,63)(27,55)(28,56)(29,66)(30,58)(31,68)(32,60)(33,70)(34,62)(35,72)(36,64)(73,122)(74,139)(75,124)(76,141)(77,126)(78,143)(79,110)(80,127)(81,112)(82,129)(83,114)(84,131)(85,116)(86,133)(87,118)(88,135)(89,120)(90,137)(91,119)(92,136)(93,121)(94,138)(95,123)(96,140)(97,125)(98,142)(99,109)(100,144)(101,111)(102,128)(103,113)(104,130)(105,115)(106,132)(107,117)(108,134), (1,26)(2,27)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,28)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133), (1,16)(2,17)(3,18)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,91)(89,92)(90,93)(109,143)(110,144)(111,127)(112,128)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136)(121,137)(122,138)(123,139)(124,140)(125,141)(126,142), (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), (1,103,35,73)(2,93,36,81)(3,101,28,89)(4,91,29,79)(5,99,30,87)(6,107,31,77)(7,97,32,85)(8,105,33,75)(9,95,34,83)(10,88,20,100)(11,78,21,108)(12,86,22,98)(13,76,23,106)(14,84,24,96)(15,74,25,104)(16,82,26,94)(17,90,27,102)(18,80,19,92)(37,137,64,128)(38,111,65,120)(39,135,66,144)(40,109,67,118)(41,133,68,142)(42,125,69,116)(43,131,70,140)(44,123,71,114)(45,129,72,138)(46,121,55,112)(47,127,56,136)(48,119,57,110)(49,143,58,134)(50,117,59,126)(51,141,60,132)(52,115,61,124)(53,139,62,130)(54,113,63,122) );
G=PermutationGroup([(1,45),(2,37),(3,47),(4,39),(5,49),(6,41),(7,51),(8,43),(9,53),(10,48),(11,40),(12,50),(13,42),(14,52),(15,44),(16,54),(17,46),(18,38),(19,65),(20,57),(21,67),(22,59),(23,69),(24,61),(25,71),(26,63),(27,55),(28,56),(29,66),(30,58),(31,68),(32,60),(33,70),(34,62),(35,72),(36,64),(73,122),(74,139),(75,124),(76,141),(77,126),(78,143),(79,110),(80,127),(81,112),(82,129),(83,114),(84,131),(85,116),(86,133),(87,118),(88,135),(89,120),(90,137),(91,119),(92,136),(93,121),(94,138),(95,123),(96,140),(97,125),(98,142),(99,109),(100,144),(101,111),(102,128),(103,113),(104,130),(105,115),(106,132),(107,117),(108,134)], [(1,26),(2,27),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,28),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(73,82),(74,83),(75,84),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90),(91,100),(92,101),(93,102),(94,103),(95,104),(96,105),(97,106),(98,107),(99,108),(109,134),(110,135),(111,136),(112,137),(113,138),(114,139),(115,140),(116,141),(117,142),(118,143),(119,144),(120,127),(121,128),(122,129),(123,130),(124,131),(125,132),(126,133)], [(1,16),(2,17),(3,18),(4,10),(5,11),(6,12),(7,13),(8,14),(9,15),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(85,106),(86,107),(87,108),(88,91),(89,92),(90,93),(109,143),(110,144),(111,127),(112,128),(113,129),(114,130),(115,131),(116,132),(117,133),(118,134),(119,135),(120,136),(121,137),(122,138),(123,139),(124,140),(125,141),(126,142)], [(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)], [(1,103,35,73),(2,93,36,81),(3,101,28,89),(4,91,29,79),(5,99,30,87),(6,107,31,77),(7,97,32,85),(8,105,33,75),(9,95,34,83),(10,88,20,100),(11,78,21,108),(12,86,22,98),(13,76,23,106),(14,84,24,96),(15,74,25,104),(16,82,26,94),(17,90,27,102),(18,80,19,92),(37,137,64,128),(38,111,65,120),(39,135,66,144),(40,109,67,118),(41,133,68,142),(42,125,69,116),(43,131,70,140),(44,123,71,114),(45,129,72,138),(46,121,55,112),(47,127,56,136),(48,119,57,110),(49,143,58,134),(50,117,59,126),(51,141,60,132),(52,115,61,124),(53,139,62,130),(54,113,63,122)])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 12A | 12B | 18A | ··· | 18I | 18J | ··· | 18U | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 18 | 18 | 18 | 18 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | D9 | C3⋊D4 | D18 | D18 | C9⋊D4 | D4⋊2S3 | D4⋊2D9 |
| kernel | C23.23D18 | Dic9⋊C4 | C18.D4 | C22×Dic9 | D4×C18 | C6×D4 | C2×C18 | C2×C12 | C22×C6 | C18 | C2×D4 | C2×C6 | C2×C4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 2 | 3 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 3 | 4 | 3 | 6 | 12 | 2 | 6 |
Matrix representation of C23.23D18 ►in GL6(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 9 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 6 | 0 | 0 |
| 0 | 0 | 31 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 24 | 36 |
| 36 | 29 | 0 | 0 | 0 | 0 |
| 28 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 27 | 0 | 0 |
| 0 | 0 | 32 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 4 | 6 |
G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,3,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,9,0,0,0,0,0,36,0,0,0,0,0,0,26,31,0,0,0,0,6,20,0,0,0,0,0,0,1,24,0,0,0,0,0,36],[36,28,0,0,0,0,29,1,0,0,0,0,0,0,32,32,0,0,0,0,27,5,0,0,0,0,0,0,31,4,0,0,0,0,0,6] >;
C23.23D18 in GAP, Magma, Sage, TeX
C_2^3._{23}D_{18} % in TeX
G:=Group("C2^3.23D18"); // GroupNames label
G:=SmallGroup(288,145);
// by ID
G=gap.SmallGroup(288,145);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,219,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^18=1,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,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