direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4⋊6D10, C30.75C24, C60.210C23, C15⋊102+ (1+4), D4⋊6(C6×D5), (D4×D5)⋊4C6, C4○D20⋊5C6, D20⋊8(C2×C6), (D4×C10)⋊7C6, (C6×D4)⋊16D5, C23⋊2(C6×D5), (D4×C30)⋊14C2, (C3×D4)⋊28D10, D4⋊2D5⋊4C6, (C2×C12)⋊22D10, (C22×C6)⋊2D10, (C2×C60)⋊22C22, Dic10⋊8(C2×C6), C10.7(C23×C6), C6.75(C23×D5), C5⋊1(C3×2+ (1+4)), (D5×C12)⋊14C22, (C3×D20)⋊38C22, (D4×C15)⋊31C22, C20.21(C22×C6), (C6×D5).54C23, D10.3(C22×C6), (C2×C30).254C23, (C22×C30)⋊14C22, (C6×Dic5)⋊21C22, C12.210(C22×D5), Dic5.4(C22×C6), (C3×Dic10)⋊35C22, (C3×Dic5).56C23, (C3×D4×D5)⋊11C2, (C2×C4)⋊3(C6×D5), C4.21(D5×C2×C6), (C2×C20)⋊3(C2×C6), (C2×D4)⋊7(C3×D5), (C5×D4)⋊7(C2×C6), (C4×D5)⋊1(C2×C6), C5⋊D4⋊3(C2×C6), (C6×C5⋊D4)⋊26C2, (C2×C5⋊D4)⋊11C6, C22.6(D5×C2×C6), C2.8(D5×C22×C6), (D5×C2×C6)⋊15C22, (C3×C4○D20)⋊15C2, (C22×C10)⋊6(C2×C6), (C2×Dic5)⋊4(C2×C6), (C22×D5)⋊3(C2×C6), (C3×D4⋊2D5)⋊11C2, (C3×C5⋊D4)⋊22C22, (C2×C10).2(C22×C6), (C2×C6).21(C22×D5), SmallGroup(480,1141)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1040 in 332 conjugacy classes, 170 normal (22 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], C5, C6, C6 [×9], C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], D5 [×4], C10, C10 [×5], C12 [×2], C12 [×4], C2×C6, C2×C6 [×4], C2×C6 [×10], C15, C2×D4, C2×D4 [×8], C4○D4 [×6], Dic5 [×4], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×C10 [×4], C2×C10 [×2], C2×C12, C2×C12 [×8], C3×D4 [×4], C3×D4 [×14], C3×Q8 [×2], C22×C6 [×2], C22×C6 [×4], C3×D5 [×4], C30, C30 [×5], 2+ (1+4), Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20, C5×D4 [×4], C22×D5 [×4], C22×C10 [×2], C6×D4, C6×D4 [×8], C3×C4○D4 [×6], C3×Dic5 [×4], C60 [×2], C6×D5 [×4], C6×D5 [×4], C2×C30, C2×C30 [×4], C2×C30 [×2], C4○D20 [×2], D4×D5 [×4], D4⋊2D5 [×4], C2×C5⋊D4 [×4], D4×C10, C3×2+ (1+4), C3×Dic10 [×2], D5×C12 [×4], C3×D20 [×2], C6×Dic5 [×4], C3×C5⋊D4 [×12], C2×C60, D4×C15 [×4], D5×C2×C6 [×4], C22×C30 [×2], D4⋊6D10, C3×C4○D20 [×2], C3×D4×D5 [×4], C3×D4⋊2D5 [×4], C6×C5⋊D4 [×4], D4×C30, C3×D4⋊6D10
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], D5, C2×C6 [×35], C24, D10 [×7], C22×C6 [×15], C3×D5, 2+ (1+4), C22×D5 [×7], C23×C6, C6×D5 [×7], C23×D5, C3×2+ (1+4), D5×C2×C6 [×7], D4⋊6D10, D5×C22×C6, C3×D4⋊6D10
Generators and relations
G = < a,b,c,d,e | a3=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >
►On 120 points
(1 22 15)(2 23 11)(3 24 12)(4 25 13)(5 21 14)(6 28 17)(7 29 18)(8 30 19)(9 26 20)(10 27 16)(31 55 43)(32 51 44)(33 52 45)(34 53 41)(35 54 42)(36 59 47)(37 60 48)(38 56 49)(39 57 50)(40 58 46)(61 87 77)(62 88 78)(63 89 79)(64 90 80)(65 81 71)(66 82 72)(67 83 73)(68 84 74)(69 85 75)(70 86 76)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)
(1 92 7 97)(2 98 8 93)(3 94 9 99)(4 100 10 95)(5 96 6 91)(11 108 19 103)(12 104 20 109)(13 110 16 105)(14 106 17 101)(15 102 18 107)(21 116 28 111)(22 112 29 117)(23 118 30 113)(24 114 26 119)(25 120 27 115)(31 62 39 67)(32 68 40 63)(33 64 36 69)(34 70 37 65)(35 66 38 61)(41 76 48 71)(42 72 49 77)(43 78 50 73)(44 74 46 79)(45 80 47 75)(51 84 58 89)(52 90 59 85)(53 86 60 81)(54 82 56 87)(55 88 57 83)
(1 61)(2 67)(3 63)(4 69)(5 65)(6 70)(7 66)(8 62)(9 68)(10 64)(11 73)(12 79)(13 75)(14 71)(15 77)(16 80)(17 76)(18 72)(19 78)(20 74)(21 81)(22 87)(23 83)(24 89)(25 85)(26 84)(27 90)(28 86)(29 82)(30 88)(31 93)(32 99)(33 95)(34 91)(35 97)(36 100)(37 96)(38 92)(39 98)(40 94)(41 101)(42 107)(43 103)(44 109)(45 105)(46 104)(47 110)(48 106)(49 102)(50 108)(51 119)(52 115)(53 111)(54 117)(55 113)(56 112)(57 118)(58 114)(59 120)(60 116)
(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 5)(2 4)(6 7)(8 10)(11 13)(14 15)(16 19)(17 18)(21 22)(23 25)(27 30)(28 29)(31 36)(32 40)(33 39)(34 38)(35 37)(41 49)(42 48)(43 47)(44 46)(45 50)(51 58)(52 57)(53 56)(54 60)(55 59)(61 70)(62 69)(63 68)(64 67)(65 66)(71 72)(73 80)(74 79)(75 78)(76 77)(81 82)(83 90)(84 89)(85 88)(86 87)(91 97)(92 96)(93 95)(98 100)(101 107)(102 106)(103 105)(108 110)(111 117)(112 116)(113 115)(118 120)
G:=sub<Sym(120)| (1,22,15)(2,23,11)(3,24,12)(4,25,13)(5,21,14)(6,28,17)(7,29,18)(8,30,19)(9,26,20)(10,27,16)(31,55,43)(32,51,44)(33,52,45)(34,53,41)(35,54,42)(36,59,47)(37,60,48)(38,56,49)(39,57,50)(40,58,46)(61,87,77)(62,88,78)(63,89,79)(64,90,80)(65,81,71)(66,82,72)(67,83,73)(68,84,74)(69,85,75)(70,86,76)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110), (1,92,7,97)(2,98,8,93)(3,94,9,99)(4,100,10,95)(5,96,6,91)(11,108,19,103)(12,104,20,109)(13,110,16,105)(14,106,17,101)(15,102,18,107)(21,116,28,111)(22,112,29,117)(23,118,30,113)(24,114,26,119)(25,120,27,115)(31,62,39,67)(32,68,40,63)(33,64,36,69)(34,70,37,65)(35,66,38,61)(41,76,48,71)(42,72,49,77)(43,78,50,73)(44,74,46,79)(45,80,47,75)(51,84,58,89)(52,90,59,85)(53,86,60,81)(54,82,56,87)(55,88,57,83), (1,61)(2,67)(3,63)(4,69)(5,65)(6,70)(7,66)(8,62)(9,68)(10,64)(11,73)(12,79)(13,75)(14,71)(15,77)(16,80)(17,76)(18,72)(19,78)(20,74)(21,81)(22,87)(23,83)(24,89)(25,85)(26,84)(27,90)(28,86)(29,82)(30,88)(31,93)(32,99)(33,95)(34,91)(35,97)(36,100)(37,96)(38,92)(39,98)(40,94)(41,101)(42,107)(43,103)(44,109)(45,105)(46,104)(47,110)(48,106)(49,102)(50,108)(51,119)(52,115)(53,111)(54,117)(55,113)(56,112)(57,118)(58,114)(59,120)(60,116), (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), (1,5)(2,4)(6,7)(8,10)(11,13)(14,15)(16,19)(17,18)(21,22)(23,25)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)(41,49)(42,48)(43,47)(44,46)(45,50)(51,58)(52,57)(53,56)(54,60)(55,59)(61,70)(62,69)(63,68)(64,67)(65,66)(71,72)(73,80)(74,79)(75,78)(76,77)(81,82)(83,90)(84,89)(85,88)(86,87)(91,97)(92,96)(93,95)(98,100)(101,107)(102,106)(103,105)(108,110)(111,117)(112,116)(113,115)(118,120)>;
G:=Group( (1,22,15)(2,23,11)(3,24,12)(4,25,13)(5,21,14)(6,28,17)(7,29,18)(8,30,19)(9,26,20)(10,27,16)(31,55,43)(32,51,44)(33,52,45)(34,53,41)(35,54,42)(36,59,47)(37,60,48)(38,56,49)(39,57,50)(40,58,46)(61,87,77)(62,88,78)(63,89,79)(64,90,80)(65,81,71)(66,82,72)(67,83,73)(68,84,74)(69,85,75)(70,86,76)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110), (1,92,7,97)(2,98,8,93)(3,94,9,99)(4,100,10,95)(5,96,6,91)(11,108,19,103)(12,104,20,109)(13,110,16,105)(14,106,17,101)(15,102,18,107)(21,116,28,111)(22,112,29,117)(23,118,30,113)(24,114,26,119)(25,120,27,115)(31,62,39,67)(32,68,40,63)(33,64,36,69)(34,70,37,65)(35,66,38,61)(41,76,48,71)(42,72,49,77)(43,78,50,73)(44,74,46,79)(45,80,47,75)(51,84,58,89)(52,90,59,85)(53,86,60,81)(54,82,56,87)(55,88,57,83), (1,61)(2,67)(3,63)(4,69)(5,65)(6,70)(7,66)(8,62)(9,68)(10,64)(11,73)(12,79)(13,75)(14,71)(15,77)(16,80)(17,76)(18,72)(19,78)(20,74)(21,81)(22,87)(23,83)(24,89)(25,85)(26,84)(27,90)(28,86)(29,82)(30,88)(31,93)(32,99)(33,95)(34,91)(35,97)(36,100)(37,96)(38,92)(39,98)(40,94)(41,101)(42,107)(43,103)(44,109)(45,105)(46,104)(47,110)(48,106)(49,102)(50,108)(51,119)(52,115)(53,111)(54,117)(55,113)(56,112)(57,118)(58,114)(59,120)(60,116), (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), (1,5)(2,4)(6,7)(8,10)(11,13)(14,15)(16,19)(17,18)(21,22)(23,25)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)(41,49)(42,48)(43,47)(44,46)(45,50)(51,58)(52,57)(53,56)(54,60)(55,59)(61,70)(62,69)(63,68)(64,67)(65,66)(71,72)(73,80)(74,79)(75,78)(76,77)(81,82)(83,90)(84,89)(85,88)(86,87)(91,97)(92,96)(93,95)(98,100)(101,107)(102,106)(103,105)(108,110)(111,117)(112,116)(113,115)(118,120) );
G=PermutationGroup([(1,22,15),(2,23,11),(3,24,12),(4,25,13),(5,21,14),(6,28,17),(7,29,18),(8,30,19),(9,26,20),(10,27,16),(31,55,43),(32,51,44),(33,52,45),(34,53,41),(35,54,42),(36,59,47),(37,60,48),(38,56,49),(39,57,50),(40,58,46),(61,87,77),(62,88,78),(63,89,79),(64,90,80),(65,81,71),(66,82,72),(67,83,73),(68,84,74),(69,85,75),(70,86,76),(91,111,101),(92,112,102),(93,113,103),(94,114,104),(95,115,105),(96,116,106),(97,117,107),(98,118,108),(99,119,109),(100,120,110)], [(1,92,7,97),(2,98,8,93),(3,94,9,99),(4,100,10,95),(5,96,6,91),(11,108,19,103),(12,104,20,109),(13,110,16,105),(14,106,17,101),(15,102,18,107),(21,116,28,111),(22,112,29,117),(23,118,30,113),(24,114,26,119),(25,120,27,115),(31,62,39,67),(32,68,40,63),(33,64,36,69),(34,70,37,65),(35,66,38,61),(41,76,48,71),(42,72,49,77),(43,78,50,73),(44,74,46,79),(45,80,47,75),(51,84,58,89),(52,90,59,85),(53,86,60,81),(54,82,56,87),(55,88,57,83)], [(1,61),(2,67),(3,63),(4,69),(5,65),(6,70),(7,66),(8,62),(9,68),(10,64),(11,73),(12,79),(13,75),(14,71),(15,77),(16,80),(17,76),(18,72),(19,78),(20,74),(21,81),(22,87),(23,83),(24,89),(25,85),(26,84),(27,90),(28,86),(29,82),(30,88),(31,93),(32,99),(33,95),(34,91),(35,97),(36,100),(37,96),(38,92),(39,98),(40,94),(41,101),(42,107),(43,103),(44,109),(45,105),(46,104),(47,110),(48,106),(49,102),(50,108),(51,119),(52,115),(53,111),(54,117),(55,113),(56,112),(57,118),(58,114),(59,120),(60,116)], [(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)], [(1,5),(2,4),(6,7),(8,10),(11,13),(14,15),(16,19),(17,18),(21,22),(23,25),(27,30),(28,29),(31,36),(32,40),(33,39),(34,38),(35,37),(41,49),(42,48),(43,47),(44,46),(45,50),(51,58),(52,57),(53,56),(54,60),(55,59),(61,70),(62,69),(63,68),(64,67),(65,66),(71,72),(73,80),(74,79),(75,78),(76,77),(81,82),(83,90),(84,89),(85,88),(86,87),(91,97),(92,96),(93,95),(98,100),(101,107),(102,106),(103,105),(108,110),(111,117),(112,116),(113,115),(118,120)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 | 0 |
| 0 | 0 | 0 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 0 | 0 | 47 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 43 | 0 | 0 | 0 | 0 |
| 17 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1,0,0,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,17,0,0,0,0,43,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1] >;
111 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 2G | 2H | 2I | 2J | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | ··· | 6L | 6M | ··· | 6T | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
111 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D5 | D10 | D10 | D10 | C3×D5 | C6×D5 | C6×D5 | C6×D5 | 2+ (1+4) | C3×2+ (1+4) | D4⋊6D10 | C3×D4⋊6D10 |
| kernel | C3×D4⋊6D10 | C3×C4○D20 | C3×D4×D5 | C3×D4⋊2D5 | C6×C5⋊D4 | D4×C30 | D4⋊6D10 | C4○D20 | D4×D5 | D4⋊2D5 | C2×C5⋊D4 | D4×C10 | C6×D4 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | C2×C4 | D4 | C23 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 2 | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 8 | 4 | 4 | 4 | 16 | 8 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_3\times D_4\rtimes_6D_{10} % in TeX
G:=Group("C3xD4:6D10"); // GroupNames label
G:=SmallGroup(480,1141);
// by ID
G=gap.SmallGroup(480,1141);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,555,1571,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^10=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations