metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊2D14, C14.252+ 1+4, (C2×D4)⋊5D14, C22≀C2⋊2D7, C22⋊C4⋊5D14, D14.2(C2×D4), (C22×D7)⋊6D4, C23⋊D14⋊3C2, C28⋊2D4⋊12C2, (D4×C14)⋊6C22, C24⋊D7⋊6C2, C7⋊2(C23⋊3D4), D14⋊D4⋊12C2, C22.40(D4×D7), D14⋊C4⋊10C22, (C2×D28)⋊18C22, (C2×C28).27C23, Dic7⋊C4⋊8C22, C4⋊Dic7⋊25C22, (C23×C14)⋊9C22, C14.55(C22×D4), (C23×D7)⋊6C22, D14.D4⋊12C2, (C2×C14).133C24, C22.D28⋊9C2, (C22×C14).8C23, C23.D7⋊14C22, C2.27(D4⋊6D14), C23.18D14⋊4C2, (C2×Dic7).60C23, C22.154(C23×D7), C23.107(C22×D7), (C22×Dic7)⋊12C22, (C22×D7).182C23, (C2×D4×D7)⋊6C2, C2.28(C2×D4×D7), (C2×C4×D7)⋊6C22, (D7×C22⋊C4)⋊2C2, (C7×C22≀C2)⋊4C2, (C2×C14).53(C2×D4), (C22×C7⋊D4)⋊7C2, (C2×C7⋊D4)⋊38C22, (C7×C22⋊C4)⋊4C22, (C2×C4).27(C22×D7), SmallGroup(448,1042)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊2D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 1932 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22≀C2, C22≀C2, C4⋊D4, C22.D4, C22×D4, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C23⋊3D4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C2×C4×D7, C2×C4×D7, C2×D28, D4×D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C2×C7⋊D4, D4×C14, D4×C14, C23×D7, C23×C14, D7×C22⋊C4, D14.D4, D14⋊D4, C22.D28, C23.18D14, C23⋊D14, C28⋊2D4, C24⋊D7, C7×C22≀C2, C2×D4×D7, C22×C7⋊D4, C24⋊2D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, C22×D7, C23⋊3D4, D4×D7, C23×D7, C2×D4×D7, D4⋊6D14, C24⋊2D14
►On 112 points
Generators in S112
(2 81)(4 83)(6 71)(8 73)(10 75)(12 77)(14 79)(16 104)(18 106)(20 108)(22 110)(24 112)(26 100)(28 102)(30 58)(32 60)(34 62)(36 64)(38 66)(40 68)(42 70)(44 95)(46 97)(48 85)(50 87)(52 89)(54 91)(56 93)
(1 15)(2 68)(3 17)(4 70)(5 19)(6 58)(7 21)(8 60)(9 23)(10 62)(11 25)(12 64)(13 27)(14 66)(16 44)(18 46)(20 48)(22 50)(24 52)(26 54)(28 56)(29 98)(30 71)(31 86)(32 73)(33 88)(34 75)(35 90)(36 77)(37 92)(38 79)(39 94)(40 81)(41 96)(42 83)(43 67)(45 69)(47 57)(49 59)(51 61)(53 63)(55 65)(72 109)(74 111)(76 99)(78 101)(80 103)(82 105)(84 107)(85 108)(87 110)(89 112)(91 100)(93 102)(95 104)(97 106)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 99)(26 100)(27 101)(28 102)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 94)(44 95)(45 96)(46 97)(47 98)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 67)(16 68)(17 69)(18 70)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)
(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)
(1 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 42)(26 41)(27 40)(28 39)(57 112)(58 111)(59 110)(60 109)(61 108)(62 107)(63 106)(64 105)(65 104)(66 103)(67 102)(68 101)(69 100)(70 99)(71 88)(72 87)(73 86)(74 85)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)
G:=sub<Sym(112)| (2,81)(4,83)(6,71)(8,73)(10,75)(12,77)(14,79)(16,104)(18,106)(20,108)(22,110)(24,112)(26,100)(28,102)(30,58)(32,60)(34,62)(36,64)(38,66)(40,68)(42,70)(44,95)(46,97)(48,85)(50,87)(52,89)(54,91)(56,93), (1,15)(2,68)(3,17)(4,70)(5,19)(6,58)(7,21)(8,60)(9,23)(10,62)(11,25)(12,64)(13,27)(14,66)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(29,98)(30,71)(31,86)(32,73)(33,88)(34,75)(35,90)(36,77)(37,92)(38,79)(39,94)(40,81)(41,96)(42,83)(43,67)(45,69)(47,57)(49,59)(51,61)(53,63)(55,65)(72,109)(74,111)(76,99)(78,101)(80,103)(82,105)(84,107)(85,108)(87,110)(89,112)(91,100)(93,102)(95,104)(97,106), (1,80)(2,81)(3,82)(4,83)(5,84)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,99)(26,100)(27,101)(28,102)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,94)(44,95)(45,96)(46,97)(47,98)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,67)(16,68)(17,69)(18,70)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,42)(26,41)(27,40)(28,39)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,88)(72,87)(73,86)(74,85)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)>;
G:=Group( (2,81)(4,83)(6,71)(8,73)(10,75)(12,77)(14,79)(16,104)(18,106)(20,108)(22,110)(24,112)(26,100)(28,102)(30,58)(32,60)(34,62)(36,64)(38,66)(40,68)(42,70)(44,95)(46,97)(48,85)(50,87)(52,89)(54,91)(56,93), (1,15)(2,68)(3,17)(4,70)(5,19)(6,58)(7,21)(8,60)(9,23)(10,62)(11,25)(12,64)(13,27)(14,66)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(29,98)(30,71)(31,86)(32,73)(33,88)(34,75)(35,90)(36,77)(37,92)(38,79)(39,94)(40,81)(41,96)(42,83)(43,67)(45,69)(47,57)(49,59)(51,61)(53,63)(55,65)(72,109)(74,111)(76,99)(78,101)(80,103)(82,105)(84,107)(85,108)(87,110)(89,112)(91,100)(93,102)(95,104)(97,106), (1,80)(2,81)(3,82)(4,83)(5,84)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,99)(26,100)(27,101)(28,102)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,94)(44,95)(45,96)(46,97)(47,98)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,67)(16,68)(17,69)(18,70)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,42)(26,41)(27,40)(28,39)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,88)(72,87)(73,86)(74,85)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89) );
G=PermutationGroup([[(2,81),(4,83),(6,71),(8,73),(10,75),(12,77),(14,79),(16,104),(18,106),(20,108),(22,110),(24,112),(26,100),(28,102),(30,58),(32,60),(34,62),(36,64),(38,66),(40,68),(42,70),(44,95),(46,97),(48,85),(50,87),(52,89),(54,91),(56,93)], [(1,15),(2,68),(3,17),(4,70),(5,19),(6,58),(7,21),(8,60),(9,23),(10,62),(11,25),(12,64),(13,27),(14,66),(16,44),(18,46),(20,48),(22,50),(24,52),(26,54),(28,56),(29,98),(30,71),(31,86),(32,73),(33,88),(34,75),(35,90),(36,77),(37,92),(38,79),(39,94),(40,81),(41,96),(42,83),(43,67),(45,69),(47,57),(49,59),(51,61),(53,63),(55,65),(72,109),(74,111),(76,99),(78,101),(80,103),(82,105),(84,107),(85,108),(87,110),(89,112),(91,100),(93,102),(95,104),(97,106)], [(1,80),(2,81),(3,82),(4,83),(5,84),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,99),(26,100),(27,101),(28,102),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(43,94),(44,95),(45,96),(46,97),(47,98),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,67),(16,68),(17,69),(18,70),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,107),(30,108),(31,109),(32,110),(33,111),(34,112),(35,99),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,106),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98)], [(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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,42),(26,41),(27,40),(28,39),(57,112),(58,111),(59,110),(60,109),(61,108),(62,107),(63,106),(64,105),(65,104),(66,103),(67,102),(68,101),(69,100),(70,99),(71,88),(72,87),(73,86),(74,85),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | ··· | 4H | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 14AB | 14AC | 14AD | 28A | ··· | 28I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | 2+ 1+4 | D4×D7 | D4⋊6D14 |
| kernel | C24⋊2D14 | D7×C22⋊C4 | D14.D4 | D14⋊D4 | C22.D28 | C23.18D14 | C23⋊D14 | C28⋊2D4 | C24⋊D7 | C7×C22≀C2 | C2×D4×D7 | C22×C7⋊D4 | C22×D7 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C14 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 9 | 9 | 3 | 2 | 6 | 12 |
Matrix representation of C24⋊2D14 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 13 | 16 | 0 | 0 | 0 | 0 |
| 4 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 2 | 0 | 0 |
| 0 | 0 | 27 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 2 |
| 0 | 0 | 0 | 0 | 27 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 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 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 25 |
| 0 | 0 | 0 | 0 | 4 | 25 |
| 0 | 0 | 11 | 25 | 0 | 0 |
| 0 | 0 | 4 | 25 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 27 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 11 |
| 0 | 0 | 0 | 0 | 25 | 4 |
| 0 | 0 | 25 | 11 | 0 | 0 |
| 0 | 0 | 25 | 4 | 0 | 0 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[13,4,0,0,0,0,16,16,0,0,0,0,0,0,18,27,0,0,0,0,2,11,0,0,0,0,0,0,18,27,0,0,0,0,2,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,28,0,0,0,0,0,0,0,0,11,4,0,0,0,0,25,25,0,0,11,4,0,0,0,0,25,25,0,0],[28,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,25,25,0,0,0,0,11,4,0,0,25,25,0,0,0,0,11,4,0,0] >;
C24⋊2D14 in GAP, Magma, Sage, TeX
C_2^4\rtimes_2D_{14} % in TeX
G:=Group("C2^4:2D14"); // GroupNames label
G:=SmallGroup(448,1042);
// by ID
G=gap.SmallGroup(448,1042);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,675,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=f^2=1,a*b=b*a,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations