direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3xD5xC4oD4, C30.79C24, C60.214C23, (D4xD5):6C6, D4:7(C6xD5), Q8:7(C6xD5), (Q8xD5):9C6, C4oD20:7C6, D20:10(C2xC6), D4:2D5:6C6, (C3xD4):29D10, (C2xC12):30D10, Q8:2D5:9C6, (C3xQ8):25D10, (C2xC60):23C22, C6.79(C23xD5), Dic10:10(C2xC6), (D4xC15):32C22, (D5xC12):26C22, (C3xD20):40C22, C20.43(C22xC6), C10.11(C23xC6), (Q8xC15):28C22, (C6xD5).57C23, (C2xC30).255C23, (C6xDic5):37C22, D10.17(C22xC6), C12.214(C22xD5), (C3xDic10):37C22, Dic5.17(C22xC6), (C3xDic5).59C23, (C2xC4xD5):6C6, C5:4(C6xC4oD4), (C3xD4xD5):13C2, (C2xC4):7(C6xD5), (C3xQ8xD5):13C2, C4.25(D5xC2xC6), (C2xC20):4(C2xC6), (C5xC4oD4):7C6, (C4xD5):7(C2xC6), (D5xC2xC12):16C2, (C5xD4):8(C2xC6), C15:30(C2xC4oD4), C5:D4:4(C2xC6), (C5xQ8):9(C2xC6), (C15xC4oD4):8C2, C22.2(D5xC2xC6), (C3xC4oD20):17C2, C2.12(D5xC22xC6), (C3xD4:2D5):13C2, (C3xQ8:2D5):13C2, (C2xDic5):10(C2xC6), (C3xC5:D4):19C22, (C2xC10).3(C22xC6), (D5xC2xC6).141C22, (C2xC6).22(C22xD5), (C22xD5).36(C2xC6), SmallGroup(480,1145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xD5xC4oD4
G = < a,b,c,d,e,f | a3=b5=c2=d4=f2=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=d2e >
Subgroups: 944 in 328 conjugacy classes, 174 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C5, C6, C6, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C12, C12, C12, C2xC6, C2xC6, C15, C22xC4, C2xD4, C2xQ8, C4oD4, C4oD4, Dic5, Dic5, C20, C20, D10, D10, D10, C2xC10, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C3xQ8, C22xC6, C3xD5, C3xD5, C30, C30, C2xC4oD4, Dic10, C4xD5, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C5xD4, C5xQ8, C22xD5, C22xC12, C6xD4, C6xQ8, C3xC4oD4, C3xC4oD4, C3xDic5, C3xDic5, C60, C60, C6xD5, C6xD5, C6xD5, C2xC30, C2xC4xD5, C4oD20, D4xD5, D4:2D5, Q8xD5, Q8:2D5, C5xC4oD4, C6xC4oD4, C3xDic10, D5xC12, D5xC12, C3xD20, C6xDic5, C3xC5:D4, C2xC60, D4xC15, Q8xC15, D5xC2xC6, D5xC4oD4, D5xC2xC12, C3xC4oD20, C3xD4xD5, C3xD4:2D5, C3xQ8xD5, C3xQ8:2D5, C15xC4oD4, C3xD5xC4oD4
Quotients: C1, C2, C3, C22, C6, C23, D5, C2xC6, C4oD4, C24, D10, C22xC6, C3xD5, C2xC4oD4, C22xD5, C3xC4oD4, C23xC6, C6xD5, C23xD5, C6xC4oD4, D5xC2xC6, D5xC4oD4, D5xC22xC6, C3xD5xC4oD4
►On 120 points
Generators in S120
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(61 81 71)(62 82 72)(63 83 73)(64 84 74)(65 85 75)(66 86 76)(67 87 77)(68 88 78)(69 89 79)(70 90 80)(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 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)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)(82 85)(83 84)(87 90)(88 89)(92 95)(93 94)(97 100)(98 99)(102 105)(103 104)(107 110)(108 109)(112 115)(113 114)(117 120)(118 119)
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 56 26 51)(22 57 27 52)(23 58 28 53)(24 59 29 54)(25 60 30 55)(61 96 66 91)(62 97 67 92)(63 98 68 93)(64 99 69 94)(65 100 70 95)(71 106 76 101)(72 107 77 102)(73 108 78 103)(74 109 79 104)(75 110 80 105)(81 116 86 111)(82 117 87 112)(83 118 88 113)(84 119 89 114)(85 120 90 115)
(1 64 9 69)(2 65 10 70)(3 61 6 66)(4 62 7 67)(5 63 8 68)(11 71 16 76)(12 72 17 77)(13 73 18 78)(14 74 19 79)(15 75 20 80)(21 81 26 86)(22 82 27 87)(23 83 28 88)(24 84 29 89)(25 85 30 90)(31 91 36 96)(32 92 37 97)(33 93 38 98)(34 94 39 99)(35 95 40 100)(41 101 46 106)(42 102 47 107)(43 103 48 108)(44 104 49 109)(45 105 50 110)(51 111 56 116)(52 112 57 117)(53 113 58 118)(54 114 59 119)(55 115 60 120)
(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)(111 116)(112 117)(113 118)(114 119)(115 120)
G:=sub<Sym(120)| (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,81,71)(62,82,72)(63,83,73)(64,84,74)(65,85,75)(66,86,76)(67,87,77)(68,88,78)(69,89,79)(70,90,80)(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), (1,2,3,4,5)(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)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119), (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55)(61,96,66,91)(62,97,67,92)(63,98,68,93)(64,99,69,94)(65,100,70,95)(71,106,76,101)(72,107,77,102)(73,108,78,103)(74,109,79,104)(75,110,80,105)(81,116,86,111)(82,117,87,112)(83,118,88,113)(84,119,89,114)(85,120,90,115), (1,64,9,69)(2,65,10,70)(3,61,6,66)(4,62,7,67)(5,63,8,68)(11,71,16,76)(12,72,17,77)(13,73,18,78)(14,74,19,79)(15,75,20,80)(21,81,26,86)(22,82,27,87)(23,83,28,88)(24,84,29,89)(25,85,30,90)(31,91,36,96)(32,92,37,97)(33,93,38,98)(34,94,39,99)(35,95,40,100)(41,101,46,106)(42,102,47,107)(43,103,48,108)(44,104,49,109)(45,105,50,110)(51,111,56,116)(52,112,57,117)(53,113,58,118)(54,114,59,119)(55,115,60,120), (61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,116)(112,117)(113,118)(114,119)(115,120)>;
G:=Group( (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,81,71)(62,82,72)(63,83,73)(64,84,74)(65,85,75)(66,86,76)(67,87,77)(68,88,78)(69,89,79)(70,90,80)(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), (1,2,3,4,5)(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)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119), (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55)(61,96,66,91)(62,97,67,92)(63,98,68,93)(64,99,69,94)(65,100,70,95)(71,106,76,101)(72,107,77,102)(73,108,78,103)(74,109,79,104)(75,110,80,105)(81,116,86,111)(82,117,87,112)(83,118,88,113)(84,119,89,114)(85,120,90,115), (1,64,9,69)(2,65,10,70)(3,61,6,66)(4,62,7,67)(5,63,8,68)(11,71,16,76)(12,72,17,77)(13,73,18,78)(14,74,19,79)(15,75,20,80)(21,81,26,86)(22,82,27,87)(23,83,28,88)(24,84,29,89)(25,85,30,90)(31,91,36,96)(32,92,37,97)(33,93,38,98)(34,94,39,99)(35,95,40,100)(41,101,46,106)(42,102,47,107)(43,103,48,108)(44,104,49,109)(45,105,50,110)(51,111,56,116)(52,112,57,117)(53,113,58,118)(54,114,59,119)(55,115,60,120), (61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,116)(112,117)(113,118)(114,119)(115,120) );
G=PermutationGroup([[(1,24,14),(2,25,15),(3,21,11),(4,22,12),(5,23,13),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50),(61,81,71),(62,82,72),(63,83,73),(64,84,74),(65,85,75),(66,86,76),(67,87,77),(68,88,78),(69,89,79),(70,90,80),(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)], [(1,2,3,4,5),(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),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39),(42,45),(43,44),(47,50),(48,49),(52,55),(53,54),(57,60),(58,59),(62,65),(63,64),(67,70),(68,69),(72,75),(73,74),(77,80),(78,79),(82,85),(83,84),(87,90),(88,89),(92,95),(93,94),(97,100),(98,99),(102,105),(103,104),(107,110),(108,109),(112,115),(113,114),(117,120),(118,119)], [(1,39,9,34),(2,40,10,35),(3,36,6,31),(4,37,7,32),(5,38,8,33),(11,46,16,41),(12,47,17,42),(13,48,18,43),(14,49,19,44),(15,50,20,45),(21,56,26,51),(22,57,27,52),(23,58,28,53),(24,59,29,54),(25,60,30,55),(61,96,66,91),(62,97,67,92),(63,98,68,93),(64,99,69,94),(65,100,70,95),(71,106,76,101),(72,107,77,102),(73,108,78,103),(74,109,79,104),(75,110,80,105),(81,116,86,111),(82,117,87,112),(83,118,88,113),(84,119,89,114),(85,120,90,115)], [(1,64,9,69),(2,65,10,70),(3,61,6,66),(4,62,7,67),(5,63,8,68),(11,71,16,76),(12,72,17,77),(13,73,18,78),(14,74,19,79),(15,75,20,80),(21,81,26,86),(22,82,27,87),(23,83,28,88),(24,84,29,89),(25,85,30,90),(31,91,36,96),(32,92,37,97),(33,93,38,98),(34,94,39,99),(35,95,40,100),(41,101,46,106),(42,102,47,107),(43,103,48,108),(44,104,49,109),(45,105,50,110),(51,111,56,116),(52,112,57,117),(53,113,58,118),(54,114,59,119),(55,115,60,120)], [(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80),(81,86),(82,87),(83,88),(84,89),(85,90),(91,96),(92,97),(93,98),(94,99),(95,100),(101,106),(102,107),(103,108),(104,109),(105,110),(111,116),(112,117),(113,118),(114,119),(115,120)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | 6M | ··· | 6R | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 12O | ··· | 12T | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 10 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | D5 | C4oD4 | D10 | D10 | D10 | C3xD5 | C3xC4oD4 | C6xD5 | C6xD5 | C6xD5 | D5xC4oD4 | C3xD5xC4oD4 |
| kernel | C3xD5xC4oD4 | D5xC2xC12 | C3xC4oD20 | C3xD4xD5 | C3xD4:2D5 | C3xQ8xD5 | C3xQ8:2D5 | C15xC4oD4 | D5xC4oD4 | C2xC4xD5 | C4oD20 | D4xD5 | D4:2D5 | Q8xD5 | Q8:2D5 | C5xC4oD4 | C3xC4oD4 | C3xD5 | C2xC12 | C3xD4 | C3xQ8 | C4oD4 | D5 | C2xC4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 6 | 6 | 2 | 4 | 8 | 12 | 12 | 4 | 4 | 8 |
Matrix representation of C3xD5xC4oD4 ►in GL4(F61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 47 |
| 0 | 1 | 0 | 0 |
| 60 | 43 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 2 |
| 0 | 0 | 59 | 53 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 53 | 60 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,47,0,0,0,0,47],[0,60,0,0,1,43,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,50,0,0,0,0,50],[1,0,0,0,0,1,0,0,0,0,8,59,0,0,2,53],[1,0,0,0,0,1,0,0,0,0,1,53,0,0,0,60] >;
C3xD5xC4oD4 in GAP, Magma, Sage, TeX
C_3\times D_5\times C_4\circ D_4
% in TeX
G:=Group("C3xD5xC4oD4"); // GroupNames label
G:=SmallGroup(480,1145);
// by ID
G=gap.SmallGroup(480,1145);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,268,794,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^5=c^2=d^4=f^2=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*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,d*f=f*d,f*e*f=d^2*e>;
// generators/relations