metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊C4⋊5D7, C23.5(C4×D7), (C2×Dic14)⋊4C4, C23⋊Dic7⋊3C2, C22.23(D4×D7), (C2×D4).119D14, C22⋊C4.43D14, (C22×Dic7)⋊4C4, (D4×C14).7C22, C23.1(C22×D7), (C2×Dic7).132D4, C23.1D14⋊3C2, (C22×C14).1C23, C23.D7.1C22, Dic7.3(C22⋊C4), C7⋊1(C23.C23), C23.11D14⋊23C2, (C22×Dic7).25C22, (C2×C4×D7)⋊1C4, (C2×C7⋊D4)⋊1C4, (C2×C4).5(C4×D7), (C7×C23⋊C4)⋊3C2, (C2×C28).5(C2×C4), C22.12(C2×C4×D7), (C2×C14).16(C2×D4), C2.11(D7×C22⋊C4), (C2×D4⋊2D7).1C2, C14.10(C2×C22⋊C4), (C22×C14).5(C2×C4), (C2×C14).6(C22×C4), (C2×Dic7).1(C2×C4), (C22×D7).1(C2×C4), (C2×C7⋊D4).1C22, (C7×C22⋊C4).82C22, SmallGroup(448,274)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊C4⋊5D7
G = < a,b,c,d,e,f | a2=b2=c2=d4=e7=f2=1, ab=ba, faf=ac=ca, dad-1=abc, ae=ea, dbd-1=bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=bcd, fef=e-1 >
Subgroups: 796 in 158 conjugacy classes, 51 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C23⋊C4, C23⋊C4, C42⋊C2, C2×C4○D4, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C23.C23, C4×Dic7, Dic7⋊C4, C23.D7, C7×C22⋊C4, C2×Dic14, C2×C4×D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, D4×C14, C23.1D14, C23⋊Dic7, C7×C23⋊C4, C23.11D14, C2×D4⋊2D7, C23⋊C4⋊5D7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, C22×D7, C23.C23, C2×C4×D7, D4×D7, D7×C22⋊C4, C23⋊C4⋊5D7
►On 112 points
Generators in S112
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)
(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 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 85)(8 96)(9 95)(10 94)(11 93)(12 92)(13 98)(14 97)(15 103)(16 102)(17 101)(18 100)(19 99)(20 105)(21 104)(22 110)(23 109)(24 108)(25 107)(26 106)(27 112)(28 111)(29 68)(30 67)(31 66)(32 65)(33 64)(34 70)(35 69)(36 61)(37 60)(38 59)(39 58)(40 57)(41 63)(42 62)(43 82)(44 81)(45 80)(46 79)(47 78)(48 84)(49 83)(50 75)(51 74)(52 73)(53 72)(54 71)(55 77)(56 76)
G:=sub<Sym(112)| (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (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), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,96)(9,95)(10,94)(11,93)(12,92)(13,98)(14,97)(15,103)(16,102)(17,101)(18,100)(19,99)(20,105)(21,104)(22,110)(23,109)(24,108)(25,107)(26,106)(27,112)(28,111)(29,68)(30,67)(31,66)(32,65)(33,64)(34,70)(35,69)(36,61)(37,60)(38,59)(39,58)(40,57)(41,63)(42,62)(43,82)(44,81)(45,80)(46,79)(47,78)(48,84)(49,83)(50,75)(51,74)(52,73)(53,72)(54,71)(55,77)(56,76)>;
G:=Group( (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (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), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,96)(9,95)(10,94)(11,93)(12,92)(13,98)(14,97)(15,103)(16,102)(17,101)(18,100)(19,99)(20,105)(21,104)(22,110)(23,109)(24,108)(25,107)(26,106)(27,112)(28,111)(29,68)(30,67)(31,66)(32,65)(33,64)(34,70)(35,69)(36,61)(37,60)(38,59)(39,58)(40,57)(41,63)(42,62)(43,82)(44,81)(45,80)(46,79)(47,78)(48,84)(49,83)(50,75)(51,74)(52,73)(53,72)(54,71)(55,77)(56,76) );
G=PermutationGroup([[(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112)], [(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,43,36,50),(30,44,37,51),(31,45,38,52),(32,46,39,53),(33,47,40,54),(34,48,41,55),(35,49,42,56),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,99,92,106),(86,100,93,107),(87,101,94,108),(88,102,95,109),(89,103,96,110),(90,104,97,111),(91,105,98,112)], [(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)], [(1,91),(2,90),(3,89),(4,88),(5,87),(6,86),(7,85),(8,96),(9,95),(10,94),(11,93),(12,92),(13,98),(14,97),(15,103),(16,102),(17,101),(18,100),(19,99),(20,105),(21,104),(22,110),(23,109),(24,108),(25,107),(26,106),(27,112),(28,111),(29,68),(30,67),(31,66),(32,65),(33,64),(34,70),(35,69),(36,61),(37,60),(38,59),(39,58),(40,57),(41,63),(42,62),(43,82),(44,81),(45,80),(46,79),(47,78),(48,84),(49,83),(50,75),(51,74),(52,73),(53,72),(54,71),(55,77),(56,76)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 7A | 7B | 7C | 14A | 14B | 14C | 14D | ··· | 14L | 14M | 14N | 14O | 28A | ··· | 28O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 28 | 4 | ··· | 4 | 7 | 7 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D7 | D14 | D14 | C4×D7 | C4×D7 | C23.C23 | D4×D7 | C23⋊C4⋊5D7 |
| kernel | C23⋊C4⋊5D7 | C23.1D14 | C23⋊Dic7 | C7×C23⋊C4 | C23.11D14 | C2×D4⋊2D7 | C2×Dic14 | C2×C4×D7 | C22×Dic7 | C2×C7⋊D4 | C2×Dic7 | C23⋊C4 | C22⋊C4 | C2×D4 | C2×C4 | C23 | C7 | C22 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 3 | 6 | 3 | 6 | 6 | 2 | 6 | 3 |
Matrix representation of C23⋊C4⋊5D7 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 28 | 28 | 28 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 28 | 28 | 28 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 11 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 28 | 18 | 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 |
| 18 | 28 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 11 | 17 | 17 | 17 |
| 0 | 0 | 0 | 12 | 0 | 0 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,13,0,0,0,0,0,28,1,0,0,0,0,28,0,0,0,0,1,28,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,0,0,0,0,28,0,0,0,0,0,28,0,1,0,0,0,28,1,0],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,11,28,0,0,0,0,0,0,0,1,0,0,11,0,28,0],[0,28,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,4,0,0,0,0,28,11,0,0,0,0,0,0,12,0,11,0,0,0,0,0,17,12,0,0,16,0,17,0,0,0,16,17,17,0] >;
C23⋊C4⋊5D7 in GAP, Magma, Sage, TeX
C_2^3\rtimes C_4\rtimes_5D_7
% in TeX
G:=Group("C2^3:C4:5D7"); // GroupNames label
G:=SmallGroup(448,274);
// by ID
G=gap.SmallGroup(448,274);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,219,58,570,438,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^4=e^7=f^2=1,a*b=b*a,f*a*f=a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=b*c*d,f*e*f=e^-1>;
// generators/relations