metabelian, supersoluble, monomial
Aliases: C62.221C23, (C2×C12).204D6, C62.62(C2×C4), (C22×C6).83D6, C62⋊5C4.4C2, C6.91(D4⋊2S3), (C6×C12).250C22, C6.Dic6⋊18C2, (C2×C62).60C22, C32⋊14(C42⋊C2), C2.1(C12.D6), C3⋊4(C23.16D6), C6.65(S3×C2×C4), (C2×C6).21(C4×S3), C22.6(C4×C3⋊S3), (C4×C3⋊Dic3)⋊20C2, (C2×C3⋊Dic3)⋊10C4, C23.16(C2×C3⋊S3), C22⋊C4.3(C3⋊S3), (C3×C22⋊C4).14S3, C3⋊Dic3.47(C2×C4), (C3×C6).96(C22×C4), (C3×C6).141(C4○D4), (C2×C6).238(C22×S3), (C32×C22⋊C4).6C2, C22.12(C22×C3⋊S3), (C22×C3⋊Dic3).9C2, (C2×C3⋊Dic3).153C22, C2.7(C2×C4×C3⋊S3), (C2×C4).25(C2×C3⋊S3), SmallGroup(288,734)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C62.221C23 |
Generators and relations for C62.221C23
G = < a,b,c,d,e | a6=b6=d2=1, c2=b3, e2=a3, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b3d >
Subgroups: 644 in 228 conjugacy classes, 85 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×8], C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×8], C23, C32, Dic3 [×24], C12 [×8], C2×C6 [×12], C2×C6 [×8], C42 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×2], C22×C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×32], C2×C12 [×8], C22×C6 [×4], C42⋊C2, C3⋊Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C62, C62 [×2], C62 [×2], C4×Dic3 [×8], Dic3⋊C4 [×8], C6.D4 [×4], C3×C22⋊C4 [×4], C22×Dic3 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×6], C6×C12 [×2], C2×C62, C23.16D6 [×4], C4×C3⋊Dic3 [×2], C6.Dic6 [×2], C62⋊5C4, C32×C22⋊C4, C22×C3⋊Dic3, C62.221C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, D6 [×12], C22×C4, C4○D4 [×2], C3⋊S3, C4×S3 [×8], C22×S3 [×4], C42⋊C2, C2×C3⋊S3 [×3], S3×C2×C4 [×4], D4⋊2S3 [×8], C4×C3⋊S3 [×2], C22×C3⋊S3, C23.16D6 [×4], C2×C4×C3⋊S3, C12.D6 [×2], C62.221C23
(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 31 38 57 16 52)(2 32 39 58 17 53)(3 33 40 59 18 54)(4 34 41 60 13 49)(5 35 42 55 14 50)(6 36 37 56 15 51)(7 142 27 22 123 137)(8 143 28 23 124 138)(9 144 29 24 125 133)(10 139 30 19 126 134)(11 140 25 20 121 135)(12 141 26 21 122 136)(43 93 73 63 83 71)(44 94 74 64 84 72)(45 95 75 65 79 67)(46 96 76 66 80 68)(47 91 77 61 81 69)(48 92 78 62 82 70)(85 99 112 104 132 118)(86 100 113 105 127 119)(87 101 114 106 128 120)(88 102 109 107 129 115)(89 97 110 108 130 116)(90 98 111 103 131 117)
(1 100 57 127)(2 99 58 132)(3 98 59 131)(4 97 60 130)(5 102 55 129)(6 101 56 128)(7 94 22 84)(8 93 23 83)(9 92 24 82)(10 91 19 81)(11 96 20 80)(12 95 21 79)(13 108 34 89)(14 107 35 88)(15 106 36 87)(16 105 31 86)(17 104 32 85)(18 103 33 90)(25 68 135 76)(26 67 136 75)(27 72 137 74)(28 71 138 73)(29 70 133 78)(30 69 134 77)(37 120 51 114)(38 119 52 113)(39 118 53 112)(40 117 54 111)(41 116 49 110)(42 115 50 109)(43 124 63 143)(44 123 64 142)(45 122 65 141)(46 121 66 140)(47 126 61 139)(48 125 62 144)
(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)(25 135)(26 136)(27 137)(28 138)(29 133)(30 134)(43 63)(44 64)(45 65)(46 66)(47 61)(48 62)(67 75)(68 76)(69 77)(70 78)(71 73)(72 74)(79 95)(80 96)(81 91)(82 92)(83 93)(84 94)(121 140)(122 141)(123 142)(124 143)(125 144)(126 139)
(1 70 4 67)(2 71 5 68)(3 72 6 69)(7 114 10 111)(8 109 11 112)(9 110 12 113)(13 65 16 62)(14 66 17 63)(15 61 18 64)(19 117 22 120)(20 118 23 115)(21 119 24 116)(25 132 28 129)(26 127 29 130)(27 128 30 131)(31 48 34 45)(32 43 35 46)(33 44 36 47)(37 91 40 94)(38 92 41 95)(39 93 42 96)(49 79 52 82)(50 80 53 83)(51 81 54 84)(55 76 58 73)(56 77 59 74)(57 78 60 75)(85 124 88 121)(86 125 89 122)(87 126 90 123)(97 136 100 133)(98 137 101 134)(99 138 102 135)(103 142 106 139)(104 143 107 140)(105 144 108 141)
G:=sub<Sym(144)| (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), (1,31,38,57,16,52)(2,32,39,58,17,53)(3,33,40,59,18,54)(4,34,41,60,13,49)(5,35,42,55,14,50)(6,36,37,56,15,51)(7,142,27,22,123,137)(8,143,28,23,124,138)(9,144,29,24,125,133)(10,139,30,19,126,134)(11,140,25,20,121,135)(12,141,26,21,122,136)(43,93,73,63,83,71)(44,94,74,64,84,72)(45,95,75,65,79,67)(46,96,76,66,80,68)(47,91,77,61,81,69)(48,92,78,62,82,70)(85,99,112,104,132,118)(86,100,113,105,127,119)(87,101,114,106,128,120)(88,102,109,107,129,115)(89,97,110,108,130,116)(90,98,111,103,131,117), (1,100,57,127)(2,99,58,132)(3,98,59,131)(4,97,60,130)(5,102,55,129)(6,101,56,128)(7,94,22,84)(8,93,23,83)(9,92,24,82)(10,91,19,81)(11,96,20,80)(12,95,21,79)(13,108,34,89)(14,107,35,88)(15,106,36,87)(16,105,31,86)(17,104,32,85)(18,103,33,90)(25,68,135,76)(26,67,136,75)(27,72,137,74)(28,71,138,73)(29,70,133,78)(30,69,134,77)(37,120,51,114)(38,119,52,113)(39,118,53,112)(40,117,54,111)(41,116,49,110)(42,115,50,109)(43,124,63,143)(44,123,64,142)(45,122,65,141)(46,121,66,140)(47,126,61,139)(48,125,62,144), (7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(25,135)(26,136)(27,137)(28,138)(29,133)(30,134)(43,63)(44,64)(45,65)(46,66)(47,61)(48,62)(67,75)(68,76)(69,77)(70,78)(71,73)(72,74)(79,95)(80,96)(81,91)(82,92)(83,93)(84,94)(121,140)(122,141)(123,142)(124,143)(125,144)(126,139), (1,70,4,67)(2,71,5,68)(3,72,6,69)(7,114,10,111)(8,109,11,112)(9,110,12,113)(13,65,16,62)(14,66,17,63)(15,61,18,64)(19,117,22,120)(20,118,23,115)(21,119,24,116)(25,132,28,129)(26,127,29,130)(27,128,30,131)(31,48,34,45)(32,43,35,46)(33,44,36,47)(37,91,40,94)(38,92,41,95)(39,93,42,96)(49,79,52,82)(50,80,53,83)(51,81,54,84)(55,76,58,73)(56,77,59,74)(57,78,60,75)(85,124,88,121)(86,125,89,122)(87,126,90,123)(97,136,100,133)(98,137,101,134)(99,138,102,135)(103,142,106,139)(104,143,107,140)(105,144,108,141)>;
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), (1,31,38,57,16,52)(2,32,39,58,17,53)(3,33,40,59,18,54)(4,34,41,60,13,49)(5,35,42,55,14,50)(6,36,37,56,15,51)(7,142,27,22,123,137)(8,143,28,23,124,138)(9,144,29,24,125,133)(10,139,30,19,126,134)(11,140,25,20,121,135)(12,141,26,21,122,136)(43,93,73,63,83,71)(44,94,74,64,84,72)(45,95,75,65,79,67)(46,96,76,66,80,68)(47,91,77,61,81,69)(48,92,78,62,82,70)(85,99,112,104,132,118)(86,100,113,105,127,119)(87,101,114,106,128,120)(88,102,109,107,129,115)(89,97,110,108,130,116)(90,98,111,103,131,117), (1,100,57,127)(2,99,58,132)(3,98,59,131)(4,97,60,130)(5,102,55,129)(6,101,56,128)(7,94,22,84)(8,93,23,83)(9,92,24,82)(10,91,19,81)(11,96,20,80)(12,95,21,79)(13,108,34,89)(14,107,35,88)(15,106,36,87)(16,105,31,86)(17,104,32,85)(18,103,33,90)(25,68,135,76)(26,67,136,75)(27,72,137,74)(28,71,138,73)(29,70,133,78)(30,69,134,77)(37,120,51,114)(38,119,52,113)(39,118,53,112)(40,117,54,111)(41,116,49,110)(42,115,50,109)(43,124,63,143)(44,123,64,142)(45,122,65,141)(46,121,66,140)(47,126,61,139)(48,125,62,144), (7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(25,135)(26,136)(27,137)(28,138)(29,133)(30,134)(43,63)(44,64)(45,65)(46,66)(47,61)(48,62)(67,75)(68,76)(69,77)(70,78)(71,73)(72,74)(79,95)(80,96)(81,91)(82,92)(83,93)(84,94)(121,140)(122,141)(123,142)(124,143)(125,144)(126,139), (1,70,4,67)(2,71,5,68)(3,72,6,69)(7,114,10,111)(8,109,11,112)(9,110,12,113)(13,65,16,62)(14,66,17,63)(15,61,18,64)(19,117,22,120)(20,118,23,115)(21,119,24,116)(25,132,28,129)(26,127,29,130)(27,128,30,131)(31,48,34,45)(32,43,35,46)(33,44,36,47)(37,91,40,94)(38,92,41,95)(39,93,42,96)(49,79,52,82)(50,80,53,83)(51,81,54,84)(55,76,58,73)(56,77,59,74)(57,78,60,75)(85,124,88,121)(86,125,89,122)(87,126,90,123)(97,136,100,133)(98,137,101,134)(99,138,102,135)(103,142,106,139)(104,143,107,140)(105,144,108,141) );
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)], [(1,31,38,57,16,52),(2,32,39,58,17,53),(3,33,40,59,18,54),(4,34,41,60,13,49),(5,35,42,55,14,50),(6,36,37,56,15,51),(7,142,27,22,123,137),(8,143,28,23,124,138),(9,144,29,24,125,133),(10,139,30,19,126,134),(11,140,25,20,121,135),(12,141,26,21,122,136),(43,93,73,63,83,71),(44,94,74,64,84,72),(45,95,75,65,79,67),(46,96,76,66,80,68),(47,91,77,61,81,69),(48,92,78,62,82,70),(85,99,112,104,132,118),(86,100,113,105,127,119),(87,101,114,106,128,120),(88,102,109,107,129,115),(89,97,110,108,130,116),(90,98,111,103,131,117)], [(1,100,57,127),(2,99,58,132),(3,98,59,131),(4,97,60,130),(5,102,55,129),(6,101,56,128),(7,94,22,84),(8,93,23,83),(9,92,24,82),(10,91,19,81),(11,96,20,80),(12,95,21,79),(13,108,34,89),(14,107,35,88),(15,106,36,87),(16,105,31,86),(17,104,32,85),(18,103,33,90),(25,68,135,76),(26,67,136,75),(27,72,137,74),(28,71,138,73),(29,70,133,78),(30,69,134,77),(37,120,51,114),(38,119,52,113),(39,118,53,112),(40,117,54,111),(41,116,49,110),(42,115,50,109),(43,124,63,143),(44,123,64,142),(45,122,65,141),(46,121,66,140),(47,126,61,139),(48,125,62,144)], [(7,22),(8,23),(9,24),(10,19),(11,20),(12,21),(25,135),(26,136),(27,137),(28,138),(29,133),(30,134),(43,63),(44,64),(45,65),(46,66),(47,61),(48,62),(67,75),(68,76),(69,77),(70,78),(71,73),(72,74),(79,95),(80,96),(81,91),(82,92),(83,93),(84,94),(121,140),(122,141),(123,142),(124,143),(125,144),(126,139)], [(1,70,4,67),(2,71,5,68),(3,72,6,69),(7,114,10,111),(8,109,11,112),(9,110,12,113),(13,65,16,62),(14,66,17,63),(15,61,18,64),(19,117,22,120),(20,118,23,115),(21,119,24,116),(25,132,28,129),(26,127,29,130),(27,128,30,131),(31,48,34,45),(32,43,35,46),(33,44,36,47),(37,91,40,94),(38,92,41,95),(39,93,42,96),(49,79,52,82),(50,80,53,83),(51,81,54,84),(55,76,58,73),(56,77,59,74),(57,78,60,75),(85,124,88,121),(86,125,89,122),(87,126,90,123),(97,136,100,133),(98,137,101,134),(99,138,102,135),(103,142,106,139),(104,143,107,140),(105,144,108,141)])
60 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | ··· | 6L | 6M | ··· | 6T | 12A | ··· | 12P |
order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
type | + | + | + | + | + | + | + | + | + | - | |||
image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | C4○D4 | C4×S3 | D4⋊2S3 |
kernel | C62.221C23 | C4×C3⋊Dic3 | C6.Dic6 | C62⋊5C4 | C32×C22⋊C4 | C22×C3⋊Dic3 | C2×C3⋊Dic3 | C3×C22⋊C4 | C2×C12 | C22×C6 | C3×C6 | C2×C6 | C6 |
# reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 8 | 4 | 4 | 16 | 8 |
Matrix representation of C62.221C23 ►in GL8(𝔽13)
12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 |
8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 7 | 3 |
0 | 0 | 0 | 0 | 0 | 0 | 10 | 6 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 8 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 8 | 11 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 5 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6],[1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,12,0,0,0,0,0,0,11,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;
C62.221C23 in GAP, Magma, Sage, TeX
C_6^2._{221}C_2^3
% in TeX
G:=Group("C6^2.221C2^3");
// GroupNames label
G:=SmallGroup(288,734);
// by ID
G=gap.SmallGroup(288,734);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,219,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=d^2=1,c^2=b^3,e^2=a^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,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^3*d>;
// generators/relations