direct product, metabelian, soluble, monomial, A-group
Aliases: C2×A4×Dic5, C10⋊2(C4×A4), (C10×A4)⋊4C4, C24.(C3×D5), (C23×C10).C6, C22⋊(C6×Dic5), (C23×Dic5)⋊C3, C22.9(D5×A4), (C22×C10)⋊2C12, (C2×A4).16D10, (C22×A4).2D5, C23⋊2(C3×Dic5), C23.13(C6×D5), C10.10(C22×A4), (C22×Dic5)⋊3C6, (C10×A4).16C22, C5⋊3(C2×C4×A4), C2.2(C2×D5×A4), (A4×C2×C10).2C2, (C5×A4)⋊11(C2×C4), (C2×C10)⋊4(C2×C12), (C2×C10).14(C2×A4), (C22×C10).4(C2×C6), SmallGroup(480,1044)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C10 — C2×A4×Dic5 |
Generators and relations for C2×A4×Dic5
G = < a,b,c,d,e,f | a2=b2=c2=d3=e10=1, f2=e5, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=bc=cb, be=eb, bf=fb, dcd-1=b, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 580 in 132 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C2×C4, C23, C23, C23, C10, C10, C10, C12, A4, C2×C6, C15, C22×C4, C24, Dic5, Dic5, C2×C10, C2×C10, C2×C12, C2×A4, C2×A4, C30, C23×C4, C2×Dic5, C2×Dic5, C22×C10, C22×C10, C22×C10, C4×A4, C22×A4, C3×Dic5, C5×A4, C2×C30, C22×Dic5, C22×Dic5, C23×C10, C2×C4×A4, C6×Dic5, C10×A4, C10×A4, C23×Dic5, A4×Dic5, A4×C2×C10, C2×A4×Dic5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D5, C12, A4, C2×C6, Dic5, D10, C2×C12, C2×A4, C3×D5, C2×Dic5, C4×A4, C22×A4, C3×Dic5, C6×D5, C2×C4×A4, C6×Dic5, D5×A4, A4×Dic5, C2×D5×A4, C2×A4×Dic5
►On 120 points
Generators in S120
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 111)(18 112)(19 113)(20 114)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 41)(40 42)(51 70)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(71 88)(72 89)(73 90)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(91 108)(92 109)(93 110)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 41)(40 42)(71 88)(72 89)(73 90)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(91 108)(92 109)(93 110)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 111)(18 112)(19 113)(20 114)(51 70)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(71 88)(72 89)(73 90)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)
(1 56 36)(2 57 37)(3 58 38)(4 59 39)(5 60 40)(6 51 31)(7 52 32)(8 53 33)(9 54 34)(10 55 35)(11 102 82)(12 103 83)(13 104 84)(14 105 85)(15 106 86)(16 107 87)(17 108 88)(18 109 89)(19 110 90)(20 101 81)(21 68 41)(22 69 42)(23 70 43)(24 61 44)(25 62 45)(26 63 46)(27 64 47)(28 65 48)(29 66 49)(30 67 50)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(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)
(1 75 6 80)(2 74 7 79)(3 73 8 78)(4 72 9 77)(5 71 10 76)(11 70 16 65)(12 69 17 64)(13 68 18 63)(14 67 19 62)(15 66 20 61)(21 89 26 84)(22 88 27 83)(23 87 28 82)(24 86 29 81)(25 85 30 90)(31 100 36 95)(32 99 37 94)(33 98 38 93)(34 97 39 92)(35 96 40 91)(41 109 46 104)(42 108 47 103)(43 107 48 102)(44 106 49 101)(45 105 50 110)(51 120 56 115)(52 119 57 114)(53 118 58 113)(54 117 59 112)(55 116 60 111)
G:=sub<Sym(120)| (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,111)(18,112)(19,113)(20,114)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,41)(40,42)(51,70)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(71,88)(72,89)(73,90)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(91,108)(92,109)(93,110)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107), (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,41)(40,42)(71,88)(72,89)(73,90)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(91,108)(92,109)(93,110)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107), (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,111)(18,112)(19,113)(20,114)(51,70)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(71,88)(72,89)(73,90)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87), (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,51,31)(7,52,32)(8,53,33)(9,54,34)(10,55,35)(11,102,82)(12,103,83)(13,104,84)(14,105,85)(15,106,86)(16,107,87)(17,108,88)(18,109,89)(19,110,90)(20,101,81)(21,68,41)(22,69,42)(23,70,43)(24,61,44)(25,62,45)(26,63,46)(27,64,47)(28,65,48)(29,66,49)(30,67,50)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (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), (1,75,6,80)(2,74,7,79)(3,73,8,78)(4,72,9,77)(5,71,10,76)(11,70,16,65)(12,69,17,64)(13,68,18,63)(14,67,19,62)(15,66,20,61)(21,89,26,84)(22,88,27,83)(23,87,28,82)(24,86,29,81)(25,85,30,90)(31,100,36,95)(32,99,37,94)(33,98,38,93)(34,97,39,92)(35,96,40,91)(41,109,46,104)(42,108,47,103)(43,107,48,102)(44,106,49,101)(45,105,50,110)(51,120,56,115)(52,119,57,114)(53,118,58,113)(54,117,59,112)(55,116,60,111)>;
G:=Group( (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,111)(18,112)(19,113)(20,114)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,41)(40,42)(51,70)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(71,88)(72,89)(73,90)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(91,108)(92,109)(93,110)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107), (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,41)(40,42)(71,88)(72,89)(73,90)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(91,108)(92,109)(93,110)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(100,107), (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,111)(18,112)(19,113)(20,114)(51,70)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(71,88)(72,89)(73,90)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87), (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,51,31)(7,52,32)(8,53,33)(9,54,34)(10,55,35)(11,102,82)(12,103,83)(13,104,84)(14,105,85)(15,106,86)(16,107,87)(17,108,88)(18,109,89)(19,110,90)(20,101,81)(21,68,41)(22,69,42)(23,70,43)(24,61,44)(25,62,45)(26,63,46)(27,64,47)(28,65,48)(29,66,49)(30,67,50)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (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), (1,75,6,80)(2,74,7,79)(3,73,8,78)(4,72,9,77)(5,71,10,76)(11,70,16,65)(12,69,17,64)(13,68,18,63)(14,67,19,62)(15,66,20,61)(21,89,26,84)(22,88,27,83)(23,87,28,82)(24,86,29,81)(25,85,30,90)(31,100,36,95)(32,99,37,94)(33,98,38,93)(34,97,39,92)(35,96,40,91)(41,109,46,104)(42,108,47,103)(43,107,48,102)(44,106,49,101)(45,105,50,110)(51,120,56,115)(52,119,57,114)(53,118,58,113)(54,117,59,112)(55,116,60,111) );
G=PermutationGroup([[(1,28),(2,29),(3,30),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,111),(18,112),(19,113),(20,114),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,41),(40,42),(51,70),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(71,88),(72,89),(73,90),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87),(91,108),(92,109),(93,110),(94,101),(95,102),(96,103),(97,104),(98,105),(99,106),(100,107)], [(1,28),(2,29),(3,30),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,41),(40,42),(71,88),(72,89),(73,90),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87),(91,108),(92,109),(93,110),(94,101),(95,102),(96,103),(97,104),(98,105),(99,106),(100,107)], [(1,28),(2,29),(3,30),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,111),(18,112),(19,113),(20,114),(51,70),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(71,88),(72,89),(73,90),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87)], [(1,56,36),(2,57,37),(3,58,38),(4,59,39),(5,60,40),(6,51,31),(7,52,32),(8,53,33),(9,54,34),(10,55,35),(11,102,82),(12,103,83),(13,104,84),(14,105,85),(15,106,86),(16,107,87),(17,108,88),(18,109,89),(19,110,90),(20,101,81),(21,68,41),(22,69,42),(23,70,43),(24,61,44),(25,62,45),(26,63,46),(27,64,47),(28,65,48),(29,66,49),(30,67,50),(71,111,91),(72,112,92),(73,113,93),(74,114,94),(75,115,95),(76,116,96),(77,117,97),(78,118,98),(79,119,99),(80,120,100)], [(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)], [(1,75,6,80),(2,74,7,79),(3,73,8,78),(4,72,9,77),(5,71,10,76),(11,70,16,65),(12,69,17,64),(13,68,18,63),(14,67,19,62),(15,66,20,61),(21,89,26,84),(22,88,27,83),(23,87,28,82),(24,86,29,81),(25,85,30,90),(31,100,36,95),(32,99,37,94),(33,98,38,93),(34,97,39,92),(35,96,40,91),(41,109,46,104),(42,108,47,103),(43,107,48,102),(44,106,49,101),(45,105,50,110),(51,120,56,115),(52,119,57,114),(53,118,58,113),(54,117,59,112),(55,116,60,111)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6F | 10A | ··· | 10F | 10G | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 15 | 15 | 15 | 15 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 20 | ··· | 20 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | + | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D5 | Dic5 | D10 | C3×D5 | C3×Dic5 | C6×D5 | A4 | C2×A4 | C2×A4 | C4×A4 | D5×A4 | A4×Dic5 | C2×D5×A4 |
| kernel | C2×A4×Dic5 | A4×Dic5 | A4×C2×C10 | C23×Dic5 | C10×A4 | C22×Dic5 | C23×C10 | C22×C10 | C22×A4 | C2×A4 | C2×A4 | C24 | C23 | C23 | C2×Dic5 | Dic5 | C2×C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 4 | 2 | 4 | 8 | 4 | 1 | 2 | 1 | 4 | 2 | 4 | 2 |
Matrix representation of C2×A4×Dic5 ►in GL7(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 48 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 47 | 60 |
| 47 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 48 | 47 | 59 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 |
| 44 | 60 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 37 | 53 | 0 | 0 | 0 | 0 | 0 |
| 34 | 24 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 34 | 0 | 0 | 0 |
| 0 | 0 | 8 | 20 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(7,GF(61))| [60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,48,0,0,0,0,0,60,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,47,0,0,0,0,0,0,60],[47,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,48,60,0,0,0,0,0,47,0,0,0,0,0,0,59,0,13],[44,1,0,0,0,0,0,60,0,0,0,0,0,0,0,0,17,60,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[37,34,0,0,0,0,0,53,24,0,0,0,0,0,0,0,41,8,0,0,0,0,0,34,20,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;
C2×A4×Dic5 in GAP, Magma, Sage, TeX
C_2\times A_4\times {\rm Dic}_5 % in TeX
G:=Group("C2xA4xDic5"); // GroupNames label
G:=SmallGroup(480,1044);
// by ID
G=gap.SmallGroup(480,1044);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,84,648,271,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^10=1,f^2=e^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*b*d^-1=b*c=c*b,b*e=e*b,b*f=f*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations