direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4⋊F5, Dic10⋊1C12, C15⋊12C4≀C2, D4⋊2(C3×F5), (C3×D4)⋊5F5, (C4×F5)⋊1C6, C4.F5⋊1C6, C4.2(C6×F5), (D4×C15)⋊5C4, (C5×D4)⋊2C12, (C12×F5)⋊6C2, C60.41(C2×C4), C20.2(C2×C12), D10.2(C3×D4), (C6×D5).37D4, C12.41(C2×F5), (C3×Dic10)⋊4C4, D4⋊2D5.2C6, (C3×Dic5).86D4, Dic5.21(C3×D4), C6.32(C22⋊F5), C30.32(C22⋊C4), (D5×C12).83C22, C5⋊1(C3×C4≀C2), (C3×C4.F5)⋊7C2, (C4×D5).8(C2×C6), C2.7(C3×C22⋊F5), C10.6(C3×C22⋊C4), (C3×D4⋊2D5).5C2, SmallGroup(480,288)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4⋊F5
G = < a,b,c,d,e | a3=b4=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b-1c, ede-1=d3 >
Subgroups: 328 in 88 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C15, C42, M4(2), C4○D4, Dic5, Dic5, C20, F5, D10, C2×C10, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C3×D5, C30, C30, C4≀C2, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×F5, C4×C12, C3×M4(2), C3×C4○D4, C3×Dic5, C3×Dic5, C60, C3×F5, C6×D5, C2×C30, C4.F5, C4×F5, D4⋊2D5, C3×C4≀C2, C3×C5⋊C8, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, D4×C15, C6×F5, D4⋊F5, C3×C4.F5, C12×F5, C3×D4⋊2D5, C3×D4⋊F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, F5, C2×C12, C3×D4, C4≀C2, C2×F5, C3×C22⋊C4, C3×F5, C22⋊F5, C3×C4≀C2, C6×F5, D4⋊F5, C3×C22⋊F5, C3×D4⋊F5
►On 120 points
Generators in S120
(1 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(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 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)
(1 79)(2 80)(3 76)(4 77)(5 78)(6 71)(7 72)(8 73)(9 74)(10 75)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 116)(42 117)(43 118)(44 119)(45 120)(46 111)(47 112)(48 113)(49 114)(50 115)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)
(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 14 9 19)(2 11 8 17)(3 13 7 20)(4 15 6 18)(5 12 10 16)(21 33 27 40)(22 35 26 38)(23 32 30 36)(24 34 29 39)(25 31 28 37)(41 53 47 60)(42 55 46 58)(43 52 50 56)(44 54 49 59)(45 51 48 57)(61 63 62 65)(66 68 67 70)(71 73 72 75)(76 78 77 80)(81 83 82 85)(86 88 87 90)(91 93 92 95)(96 98 97 100)(101 103 102 105)(106 108 107 110)(111 113 112 115)(116 118 117 120)
G:=sub<Sym(120)| (1,44,24)(2,45,25)(3,41,21)(4,42,22)(5,43,23)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,101,81)(62,102,82)(63,103,83)(64,104,84)(65,105,85)(66,106,86)(67,107,87)(68,108,88)(69,109,89)(70,110,90)(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), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120), (1,79)(2,80)(3,76)(4,77)(5,78)(6,71)(7,72)(8,73)(9,74)(10,75)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,96)(22,97)(23,98)(24,99)(25,100)(26,91)(27,92)(28,93)(29,94)(30,95)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,116)(42,117)(43,118)(44,119)(45,120)(46,111)(47,112)(48,113)(49,114)(50,115)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110), (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), (1,14,9,19)(2,11,8,17)(3,13,7,20)(4,15,6,18)(5,12,10,16)(21,33,27,40)(22,35,26,38)(23,32,30,36)(24,34,29,39)(25,31,28,37)(41,53,47,60)(42,55,46,58)(43,52,50,56)(44,54,49,59)(45,51,48,57)(61,63,62,65)(66,68,67,70)(71,73,72,75)(76,78,77,80)(81,83,82,85)(86,88,87,90)(91,93,92,95)(96,98,97,100)(101,103,102,105)(106,108,107,110)(111,113,112,115)(116,118,117,120)>;
G:=Group( (1,44,24)(2,45,25)(3,41,21)(4,42,22)(5,43,23)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,101,81)(62,102,82)(63,103,83)(64,104,84)(65,105,85)(66,106,86)(67,107,87)(68,108,88)(69,109,89)(70,110,90)(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), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120), (1,79)(2,80)(3,76)(4,77)(5,78)(6,71)(7,72)(8,73)(9,74)(10,75)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,96)(22,97)(23,98)(24,99)(25,100)(26,91)(27,92)(28,93)(29,94)(30,95)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,116)(42,117)(43,118)(44,119)(45,120)(46,111)(47,112)(48,113)(49,114)(50,115)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110), (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), (1,14,9,19)(2,11,8,17)(3,13,7,20)(4,15,6,18)(5,12,10,16)(21,33,27,40)(22,35,26,38)(23,32,30,36)(24,34,29,39)(25,31,28,37)(41,53,47,60)(42,55,46,58)(43,52,50,56)(44,54,49,59)(45,51,48,57)(61,63,62,65)(66,68,67,70)(71,73,72,75)(76,78,77,80)(81,83,82,85)(86,88,87,90)(91,93,92,95)(96,98,97,100)(101,103,102,105)(106,108,107,110)(111,113,112,115)(116,118,117,120) );
G=PermutationGroup([[(1,44,24),(2,45,25),(3,41,21),(4,42,22),(5,43,23),(6,46,26),(7,47,27),(8,48,28),(9,49,29),(10,50,30),(11,51,31),(12,52,32),(13,53,33),(14,54,34),(15,55,35),(16,56,36),(17,57,37),(18,58,38),(19,59,39),(20,60,40),(61,101,81),(62,102,82),(63,103,83),(64,104,84),(65,105,85),(66,106,86),(67,107,87),(68,108,88),(69,109,89),(70,110,90),(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)], [(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80),(81,91,86,96),(82,92,87,97),(83,93,88,98),(84,94,89,99),(85,95,90,100),(101,111,106,116),(102,112,107,117),(103,113,108,118),(104,114,109,119),(105,115,110,120)], [(1,79),(2,80),(3,76),(4,77),(5,78),(6,71),(7,72),(8,73),(9,74),(10,75),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,96),(22,97),(23,98),(24,99),(25,100),(26,91),(27,92),(28,93),(29,94),(30,95),(31,81),(32,82),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,116),(42,117),(43,118),(44,119),(45,120),(46,111),(47,112),(48,113),(49,114),(50,115),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106),(57,107),(58,108),(59,109),(60,110)], [(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)], [(1,14,9,19),(2,11,8,17),(3,13,7,20),(4,15,6,18),(5,12,10,16),(21,33,27,40),(22,35,26,38),(23,32,30,36),(24,34,29,39),(25,31,28,37),(41,53,47,60),(42,55,46,58),(43,52,50,56),(44,54,49,59),(45,51,48,57),(61,63,62,65),(66,68,67,70),(71,73,72,75),(76,78,77,80),(81,83,82,85),(86,88,87,90),(91,93,92,95),(96,98,97,100),(101,103,102,105),(106,108,107,110),(111,113,112,115),(116,118,117,120)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12N | 12O | 12P | 15A | 15B | 20 | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 15 | 15 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 4 | 10 | 1 | 1 | 2 | 5 | 5 | 10 | 10 | 10 | 10 | 20 | 4 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 20 | 4 | 8 | 8 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | 20 | 4 | 4 | 8 | 20 | 20 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | D4 | C3×D4 | C3×D4 | C4≀C2 | C3×C4≀C2 | F5 | C2×F5 | C3×F5 | C22⋊F5 | C6×F5 | C3×C22⋊F5 | D4⋊F5 | C3×D4⋊F5 |
| kernel | C3×D4⋊F5 | C3×C4.F5 | C12×F5 | C3×D4⋊2D5 | D4⋊F5 | C3×Dic10 | D4×C15 | C4.F5 | C4×F5 | D4⋊2D5 | Dic10 | C5×D4 | C3×Dic5 | C6×D5 | Dic5 | D10 | C15 | C5 | C3×D4 | C12 | D4 | C6 | C4 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 |
Matrix representation of C3×D4⋊F5 ►in GL8(𝔽241)
| 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 144 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 119 | 79 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 122 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 124 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 234 | 117 | 234 | 0 |
| 0 | 0 | 0 | 0 | 0 | 234 | 117 | 234 |
| 0 | 0 | 0 | 0 | 7 | 7 | 0 | 124 |
| 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 | 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 | 1 |
| 0 | 0 | 0 | 0 | 240 | 240 | 240 | 240 |
| 177 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 140 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(8,GF(241))| [15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,177,144,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,119,16,0,0,0,0,0,0,79,122,0,0,0,0,0,0,0,0,124,234,0,7,0,0,0,0,0,117,234,7,0,0,0,0,7,234,117,0,0,0,0,0,7,0,234,124],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240],[177,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,140,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,0,0,0,1,0,0,0,0,0,240,0,1] >;
C3×D4⋊F5 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes F_5
% in TeX
G:=Group("C3xD4:F5"); // GroupNames label
G:=SmallGroup(480,288);
// by ID
G=gap.SmallGroup(480,288);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,136,2524,1271,102,9414,1595]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^3>;
// generators/relations