direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C24⋊4S3, (C2×C30)⋊34D4, C24⋊5(C5×S3), C15⋊19C22≀C2, (C23×C30)⋊9C2, (C23×C10)⋊7S3, (C23×C6)⋊5C10, C6.63(D4×C10), C30.446(C2×D4), C23.30(S3×C10), C6.D4⋊13C10, (C2×C30).440C23, (C22×C10).128D6, (C10×Dic3)⋊20C22, (C22×C30).180C22, (C2×C6)⋊8(C5×D4), C3⋊3(C5×C22≀C2), (C2×C3⋊D4)⋊8C10, C22⋊4(C5×C3⋊D4), (C10×C3⋊D4)⋊23C2, (S3×C2×C10)⋊14C22, C2.26(C10×C3⋊D4), C22.66(S3×C2×C10), (C2×C10)⋊17(C3⋊D4), (C22×S3)⋊2(C2×C10), (C2×Dic3)⋊3(C2×C10), C10.148(C2×C3⋊D4), (C5×C6.D4)⋊29C2, (C2×C6).61(C22×C10), (C22×C6).42(C2×C10), (C2×C10).374(C22×S3), SmallGroup(480,832)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C24⋊4S3
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, gbg=be=eb, bf=fb, gcg=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg=f-1 >
Subgroups: 580 in 260 conjugacy classes, 82 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, C10, C10, Dic3, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C5×S3, C30, C30, C22≀C2, C2×C20, C5×D4, C22×C10, C22×C10, C6.D4, C2×C3⋊D4, C23×C6, C5×Dic3, S3×C10, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, D4×C10, C23×C10, C24⋊4S3, C10×Dic3, C5×C3⋊D4, S3×C2×C10, C22×C30, C22×C30, C5×C22≀C2, C5×C6.D4, C10×C3⋊D4, C23×C30, C5×C24⋊4S3
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C22≀C2, C5×D4, C22×C10, C2×C3⋊D4, S3×C10, D4×C10, C24⋊4S3, C5×C3⋊D4, S3×C2×C10, C5×C22≀C2, C10×C3⋊D4, C5×C24⋊4S3
►On 120 points
Generators in S120
(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 58)(2 59)(3 60)(4 56)(5 57)(6 90)(7 86)(8 87)(9 88)(10 89)(16 53)(17 54)(18 55)(19 51)(20 52)(26 66)(27 67)(28 68)(29 69)(30 70)(31 43)(32 44)(33 45)(34 41)(35 42)(61 79)(62 80)(63 76)(64 77)(65 78)
(11 111)(12 112)(13 113)(14 114)(15 115)(21 75)(22 71)(23 72)(24 73)(25 74)(36 81)(37 82)(38 83)(39 84)(40 85)(46 96)(47 97)(48 98)(49 99)(50 100)(91 110)(92 106)(93 107)(94 108)(95 109)(101 116)(102 117)(103 118)(104 119)(105 120)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 52)(7 53)(8 54)(9 55)(10 51)(11 111)(12 112)(13 113)(14 114)(15 115)(16 86)(17 87)(18 88)(19 89)(20 90)(21 75)(22 71)(23 72)(24 73)(25 74)(26 58)(27 59)(28 60)(29 56)(30 57)(31 63)(32 64)(33 65)(34 61)(35 62)(36 81)(37 82)(38 83)(39 84)(40 85)(41 79)(42 80)(43 76)(44 77)(45 78)(46 96)(47 97)(48 98)(49 99)(50 100)(91 110)(92 106)(93 107)(94 108)(95 109)(101 116)(102 117)(103 118)(104 119)(105 120)
(1 58)(2 59)(3 60)(4 56)(5 57)(6 90)(7 86)(8 87)(9 88)(10 89)(11 48)(12 49)(13 50)(14 46)(15 47)(16 53)(17 54)(18 55)(19 51)(20 52)(21 93)(22 94)(23 95)(24 91)(25 92)(26 66)(27 67)(28 68)(29 69)(30 70)(31 43)(32 44)(33 45)(34 41)(35 42)(36 103)(37 104)(38 105)(39 101)(40 102)(61 79)(62 80)(63 76)(64 77)(65 78)(71 108)(72 109)(73 110)(74 106)(75 107)(81 118)(82 119)(83 120)(84 116)(85 117)(96 114)(97 115)(98 111)(99 112)(100 113)
(1 33 90)(2 34 86)(3 35 87)(4 31 88)(5 32 89)(6 58 45)(7 59 41)(8 60 42)(9 56 43)(10 57 44)(11 36 73)(12 37 74)(13 38 75)(14 39 71)(15 40 72)(16 67 61)(17 68 62)(18 69 63)(19 70 64)(20 66 65)(21 113 83)(22 114 84)(23 115 85)(24 111 81)(25 112 82)(26 78 52)(27 79 53)(28 80 54)(29 76 55)(30 77 51)(46 101 108)(47 102 109)(48 103 110)(49 104 106)(50 105 107)(91 98 118)(92 99 119)(93 100 120)(94 96 116)(95 97 117)
(1 81)(2 82)(3 83)(4 84)(5 85)(6 91)(7 92)(8 93)(9 94)(10 95)(11 65)(12 61)(13 62)(14 63)(15 64)(16 74)(17 75)(18 71)(19 72)(20 73)(21 87)(22 88)(23 89)(24 90)(25 86)(26 103)(27 104)(28 105)(29 101)(30 102)(31 114)(32 115)(33 111)(34 112)(35 113)(36 66)(37 67)(38 68)(39 69)(40 70)(41 99)(42 100)(43 96)(44 97)(45 98)(46 76)(47 77)(48 78)(49 79)(50 80)(51 109)(52 110)(53 106)(54 107)(55 108)(56 116)(57 117)(58 118)(59 119)(60 120)
G:=sub<Sym(120)| (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), (1,58)(2,59)(3,60)(4,56)(5,57)(6,90)(7,86)(8,87)(9,88)(10,89)(16,53)(17,54)(18,55)(19,51)(20,52)(26,66)(27,67)(28,68)(29,69)(30,70)(31,43)(32,44)(33,45)(34,41)(35,42)(61,79)(62,80)(63,76)(64,77)(65,78), (11,111)(12,112)(13,113)(14,114)(15,115)(21,75)(22,71)(23,72)(24,73)(25,74)(36,81)(37,82)(38,83)(39,84)(40,85)(46,96)(47,97)(48,98)(49,99)(50,100)(91,110)(92,106)(93,107)(94,108)(95,109)(101,116)(102,117)(103,118)(104,119)(105,120), (1,66)(2,67)(3,68)(4,69)(5,70)(6,52)(7,53)(8,54)(9,55)(10,51)(11,111)(12,112)(13,113)(14,114)(15,115)(16,86)(17,87)(18,88)(19,89)(20,90)(21,75)(22,71)(23,72)(24,73)(25,74)(26,58)(27,59)(28,60)(29,56)(30,57)(31,63)(32,64)(33,65)(34,61)(35,62)(36,81)(37,82)(38,83)(39,84)(40,85)(41,79)(42,80)(43,76)(44,77)(45,78)(46,96)(47,97)(48,98)(49,99)(50,100)(91,110)(92,106)(93,107)(94,108)(95,109)(101,116)(102,117)(103,118)(104,119)(105,120), (1,58)(2,59)(3,60)(4,56)(5,57)(6,90)(7,86)(8,87)(9,88)(10,89)(11,48)(12,49)(13,50)(14,46)(15,47)(16,53)(17,54)(18,55)(19,51)(20,52)(21,93)(22,94)(23,95)(24,91)(25,92)(26,66)(27,67)(28,68)(29,69)(30,70)(31,43)(32,44)(33,45)(34,41)(35,42)(36,103)(37,104)(38,105)(39,101)(40,102)(61,79)(62,80)(63,76)(64,77)(65,78)(71,108)(72,109)(73,110)(74,106)(75,107)(81,118)(82,119)(83,120)(84,116)(85,117)(96,114)(97,115)(98,111)(99,112)(100,113), (1,33,90)(2,34,86)(3,35,87)(4,31,88)(5,32,89)(6,58,45)(7,59,41)(8,60,42)(9,56,43)(10,57,44)(11,36,73)(12,37,74)(13,38,75)(14,39,71)(15,40,72)(16,67,61)(17,68,62)(18,69,63)(19,70,64)(20,66,65)(21,113,83)(22,114,84)(23,115,85)(24,111,81)(25,112,82)(26,78,52)(27,79,53)(28,80,54)(29,76,55)(30,77,51)(46,101,108)(47,102,109)(48,103,110)(49,104,106)(50,105,107)(91,98,118)(92,99,119)(93,100,120)(94,96,116)(95,97,117), (1,81)(2,82)(3,83)(4,84)(5,85)(6,91)(7,92)(8,93)(9,94)(10,95)(11,65)(12,61)(13,62)(14,63)(15,64)(16,74)(17,75)(18,71)(19,72)(20,73)(21,87)(22,88)(23,89)(24,90)(25,86)(26,103)(27,104)(28,105)(29,101)(30,102)(31,114)(32,115)(33,111)(34,112)(35,113)(36,66)(37,67)(38,68)(39,69)(40,70)(41,99)(42,100)(43,96)(44,97)(45,98)(46,76)(47,77)(48,78)(49,79)(50,80)(51,109)(52,110)(53,106)(54,107)(55,108)(56,116)(57,117)(58,118)(59,119)(60,120)>;
G:=Group( (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), (1,58)(2,59)(3,60)(4,56)(5,57)(6,90)(7,86)(8,87)(9,88)(10,89)(16,53)(17,54)(18,55)(19,51)(20,52)(26,66)(27,67)(28,68)(29,69)(30,70)(31,43)(32,44)(33,45)(34,41)(35,42)(61,79)(62,80)(63,76)(64,77)(65,78), (11,111)(12,112)(13,113)(14,114)(15,115)(21,75)(22,71)(23,72)(24,73)(25,74)(36,81)(37,82)(38,83)(39,84)(40,85)(46,96)(47,97)(48,98)(49,99)(50,100)(91,110)(92,106)(93,107)(94,108)(95,109)(101,116)(102,117)(103,118)(104,119)(105,120), (1,66)(2,67)(3,68)(4,69)(5,70)(6,52)(7,53)(8,54)(9,55)(10,51)(11,111)(12,112)(13,113)(14,114)(15,115)(16,86)(17,87)(18,88)(19,89)(20,90)(21,75)(22,71)(23,72)(24,73)(25,74)(26,58)(27,59)(28,60)(29,56)(30,57)(31,63)(32,64)(33,65)(34,61)(35,62)(36,81)(37,82)(38,83)(39,84)(40,85)(41,79)(42,80)(43,76)(44,77)(45,78)(46,96)(47,97)(48,98)(49,99)(50,100)(91,110)(92,106)(93,107)(94,108)(95,109)(101,116)(102,117)(103,118)(104,119)(105,120), (1,58)(2,59)(3,60)(4,56)(5,57)(6,90)(7,86)(8,87)(9,88)(10,89)(11,48)(12,49)(13,50)(14,46)(15,47)(16,53)(17,54)(18,55)(19,51)(20,52)(21,93)(22,94)(23,95)(24,91)(25,92)(26,66)(27,67)(28,68)(29,69)(30,70)(31,43)(32,44)(33,45)(34,41)(35,42)(36,103)(37,104)(38,105)(39,101)(40,102)(61,79)(62,80)(63,76)(64,77)(65,78)(71,108)(72,109)(73,110)(74,106)(75,107)(81,118)(82,119)(83,120)(84,116)(85,117)(96,114)(97,115)(98,111)(99,112)(100,113), (1,33,90)(2,34,86)(3,35,87)(4,31,88)(5,32,89)(6,58,45)(7,59,41)(8,60,42)(9,56,43)(10,57,44)(11,36,73)(12,37,74)(13,38,75)(14,39,71)(15,40,72)(16,67,61)(17,68,62)(18,69,63)(19,70,64)(20,66,65)(21,113,83)(22,114,84)(23,115,85)(24,111,81)(25,112,82)(26,78,52)(27,79,53)(28,80,54)(29,76,55)(30,77,51)(46,101,108)(47,102,109)(48,103,110)(49,104,106)(50,105,107)(91,98,118)(92,99,119)(93,100,120)(94,96,116)(95,97,117), (1,81)(2,82)(3,83)(4,84)(5,85)(6,91)(7,92)(8,93)(9,94)(10,95)(11,65)(12,61)(13,62)(14,63)(15,64)(16,74)(17,75)(18,71)(19,72)(20,73)(21,87)(22,88)(23,89)(24,90)(25,86)(26,103)(27,104)(28,105)(29,101)(30,102)(31,114)(32,115)(33,111)(34,112)(35,113)(36,66)(37,67)(38,68)(39,69)(40,70)(41,99)(42,100)(43,96)(44,97)(45,98)(46,76)(47,77)(48,78)(49,79)(50,80)(51,109)(52,110)(53,106)(54,107)(55,108)(56,116)(57,117)(58,118)(59,119)(60,120) );
G=PermutationGroup([[(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)], [(1,58),(2,59),(3,60),(4,56),(5,57),(6,90),(7,86),(8,87),(9,88),(10,89),(16,53),(17,54),(18,55),(19,51),(20,52),(26,66),(27,67),(28,68),(29,69),(30,70),(31,43),(32,44),(33,45),(34,41),(35,42),(61,79),(62,80),(63,76),(64,77),(65,78)], [(11,111),(12,112),(13,113),(14,114),(15,115),(21,75),(22,71),(23,72),(24,73),(25,74),(36,81),(37,82),(38,83),(39,84),(40,85),(46,96),(47,97),(48,98),(49,99),(50,100),(91,110),(92,106),(93,107),(94,108),(95,109),(101,116),(102,117),(103,118),(104,119),(105,120)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,52),(7,53),(8,54),(9,55),(10,51),(11,111),(12,112),(13,113),(14,114),(15,115),(16,86),(17,87),(18,88),(19,89),(20,90),(21,75),(22,71),(23,72),(24,73),(25,74),(26,58),(27,59),(28,60),(29,56),(30,57),(31,63),(32,64),(33,65),(34,61),(35,62),(36,81),(37,82),(38,83),(39,84),(40,85),(41,79),(42,80),(43,76),(44,77),(45,78),(46,96),(47,97),(48,98),(49,99),(50,100),(91,110),(92,106),(93,107),(94,108),(95,109),(101,116),(102,117),(103,118),(104,119),(105,120)], [(1,58),(2,59),(3,60),(4,56),(5,57),(6,90),(7,86),(8,87),(9,88),(10,89),(11,48),(12,49),(13,50),(14,46),(15,47),(16,53),(17,54),(18,55),(19,51),(20,52),(21,93),(22,94),(23,95),(24,91),(25,92),(26,66),(27,67),(28,68),(29,69),(30,70),(31,43),(32,44),(33,45),(34,41),(35,42),(36,103),(37,104),(38,105),(39,101),(40,102),(61,79),(62,80),(63,76),(64,77),(65,78),(71,108),(72,109),(73,110),(74,106),(75,107),(81,118),(82,119),(83,120),(84,116),(85,117),(96,114),(97,115),(98,111),(99,112),(100,113)], [(1,33,90),(2,34,86),(3,35,87),(4,31,88),(5,32,89),(6,58,45),(7,59,41),(8,60,42),(9,56,43),(10,57,44),(11,36,73),(12,37,74),(13,38,75),(14,39,71),(15,40,72),(16,67,61),(17,68,62),(18,69,63),(19,70,64),(20,66,65),(21,113,83),(22,114,84),(23,115,85),(24,111,81),(25,112,82),(26,78,52),(27,79,53),(28,80,54),(29,76,55),(30,77,51),(46,101,108),(47,102,109),(48,103,110),(49,104,106),(50,105,107),(91,98,118),(92,99,119),(93,100,120),(94,96,116),(95,97,117)], [(1,81),(2,82),(3,83),(4,84),(5,85),(6,91),(7,92),(8,93),(9,94),(10,95),(11,65),(12,61),(13,62),(14,63),(15,64),(16,74),(17,75),(18,71),(19,72),(20,73),(21,87),(22,88),(23,89),(24,90),(25,86),(26,103),(27,104),(28,105),(29,101),(30,102),(31,114),(32,115),(33,111),(34,112),(35,113),(36,66),(37,67),(38,68),(39,69),(40,70),(41,99),(42,100),(43,96),(44,97),(45,98),(46,76),(47,77),(48,78),(49,79),(50,80),(51,109),(52,110),(53,106),(54,107),(55,108),(56,116),(57,117),(58,118),(59,119),(60,120)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | ··· | 6O | 10A | ··· | 10L | 10M | ··· | 10AJ | 10AK | 10AL | 10AM | 10AN | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30BH |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | 2 | 12 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D6 | C3⋊D4 | C5×S3 | C5×D4 | S3×C10 | C5×C3⋊D4 |
| kernel | C5×C24⋊4S3 | C5×C6.D4 | C10×C3⋊D4 | C23×C30 | C24⋊4S3 | C6.D4 | C2×C3⋊D4 | C23×C6 | C23×C10 | C2×C30 | C22×C10 | C2×C10 | C24 | C2×C6 | C23 | C22 |
| # reps | 1 | 3 | 3 | 1 | 4 | 12 | 12 | 4 | 1 | 6 | 3 | 12 | 4 | 24 | 12 | 48 |
Matrix representation of C5×C24⋊4S3 ►in GL4(𝔽61) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 8 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 53 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 10 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 14 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 47 | 48 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 46 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 60 |
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,8,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,53,60,0,0,0,0,1,0,0,0,10,60],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,14,47,0,0,0,0,47,0,0,0,48,13],[1,46,0,0,0,60,0,0,0,0,1,12,0,0,0,60] >;
C5×C24⋊4S3 in GAP, Magma, Sage, TeX
C_5\times C_2^4\rtimes_4S_3
% in TeX
G:=Group("C5xC2^4:4S3"); // GroupNames label
G:=SmallGroup(480,832);
// by ID
G=gap.SmallGroup(480,832);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,926,15686]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=e^2=f^3=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,g*b*g=b*e=e*b,b*f=f*b,g*c*g=c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,g*f*g=f^-1>;
// generators/relations