direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C24⋊2D5, (C2×C30)⋊32D4, (C23×C6)⋊3D5, C24⋊7(C3×D5), C15⋊18C22≀C2, (C23×C30)⋊6C2, C10.63(C6×D4), (C23×C10)⋊11C6, C30.420(C2×D4), C23.D5⋊13C6, C23.30(C6×D5), (C2×C30).378C23, (C6×Dic5)⋊20C22, (C22×C6).110D10, (C22×C30).163C22, C5⋊3(C3×C22≀C2), (C2×C5⋊D4)⋊8C6, (C2×C10)⋊11(C3×D4), (C6×C5⋊D4)⋊23C2, (D5×C2×C6)⋊14C22, C2.26(C6×C5⋊D4), C22⋊4(C3×C5⋊D4), (C2×C6)⋊13(C5⋊D4), C22.66(D5×C2×C6), (C2×Dic5)⋊3(C2×C6), (C22×D5)⋊2(C2×C6), C6.148(C2×C5⋊D4), (C3×C23.D5)⋊29C2, (C2×C10).61(C22×C6), (C22×C10).50(C2×C6), (C2×C6).374(C22×D5), SmallGroup(480,746)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C24⋊2D5
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f5=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: 736 in 260 conjugacy classes, 82 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, D10, C2×C10, C2×C10, C2×C10, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C22≀C2, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C22×C10, C3×C22⋊C4, C6×D4, C23×C6, C3×Dic5, C6×D5, C2×C30, C2×C30, C2×C30, C23.D5, C2×C5⋊D4, C23×C10, C3×C22≀C2, C6×Dic5, C3×C5⋊D4, D5×C2×C6, C22×C30, C22×C30, C24⋊2D5, C3×C23.D5, C6×C5⋊D4, C23×C30, C3×C24⋊2D5
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C22≀C2, C5⋊D4, C22×D5, C6×D4, C6×D5, C2×C5⋊D4, C3×C22≀C2, C3×C5⋊D4, D5×C2×C6, C24⋊2D5, C6×C5⋊D4, C3×C24⋊2D5
►On 120 points
Generators in S120
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(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 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)
(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(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)
(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 63)(2 62)(3 61)(4 65)(5 64)(6 68)(7 67)(8 66)(9 70)(10 69)(11 73)(12 72)(13 71)(14 75)(15 74)(16 78)(17 77)(18 76)(19 80)(20 79)(21 83)(22 82)(23 81)(24 85)(25 84)(26 88)(27 87)(28 86)(29 90)(30 89)(31 93)(32 92)(33 91)(34 95)(35 94)(36 98)(37 97)(38 96)(39 100)(40 99)(41 103)(42 102)(43 101)(44 105)(45 104)(46 108)(47 107)(48 106)(49 110)(50 109)(51 113)(52 112)(53 111)(54 115)(55 114)(56 118)(57 117)(58 116)(59 120)(60 119)
G:=sub<Sym(120)| (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(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), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60), (61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(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), (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), (1,63)(2,62)(3,61)(4,65)(5,64)(6,68)(7,67)(8,66)(9,70)(10,69)(11,73)(12,72)(13,71)(14,75)(15,74)(16,78)(17,77)(18,76)(19,80)(20,79)(21,83)(22,82)(23,81)(24,85)(25,84)(26,88)(27,87)(28,86)(29,90)(30,89)(31,93)(32,92)(33,91)(34,95)(35,94)(36,98)(37,97)(38,96)(39,100)(40,99)(41,103)(42,102)(43,101)(44,105)(45,104)(46,108)(47,107)(48,106)(49,110)(50,109)(51,113)(52,112)(53,111)(54,115)(55,114)(56,118)(57,117)(58,116)(59,120)(60,119)>;
G:=Group( (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(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), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60), (61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(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), (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), (1,63)(2,62)(3,61)(4,65)(5,64)(6,68)(7,67)(8,66)(9,70)(10,69)(11,73)(12,72)(13,71)(14,75)(15,74)(16,78)(17,77)(18,76)(19,80)(20,79)(21,83)(22,82)(23,81)(24,85)(25,84)(26,88)(27,87)(28,86)(29,90)(30,89)(31,93)(32,92)(33,91)(34,95)(35,94)(36,98)(37,97)(38,96)(39,100)(40,99)(41,103)(42,102)(43,101)(44,105)(45,104)(46,108)(47,107)(48,106)(49,110)(50,109)(51,113)(52,112)(53,111)(54,115)(55,114)(56,118)(57,117)(58,116)(59,120)(60,119) );
G=PermutationGroup([[(1,41,21),(2,42,22),(3,43,23),(4,44,24),(5,45,25),(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)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60)], [(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(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)], [(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)], [(1,63),(2,62),(3,61),(4,65),(5,64),(6,68),(7,67),(8,66),(9,70),(10,69),(11,73),(12,72),(13,71),(14,75),(15,74),(16,78),(17,77),(18,76),(19,80),(20,79),(21,83),(22,82),(23,81),(24,85),(25,84),(26,88),(27,87),(28,86),(29,90),(30,89),(31,93),(32,92),(33,91),(34,95),(35,94),(36,98),(37,97),(38,96),(39,100),(40,99),(41,103),(42,102),(43,101),(44,105),(45,104),(46,108),(47,107),(48,106),(49,110),(50,109),(51,113),(52,112),(53,111),(54,115),(55,114),(56,118),(57,117),(58,116),(59,120),(60,119)]])
138 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6R | 6S | 6T | 10A | ··· | 10AD | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 30A | ··· | 30BH |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 1 | 1 | 20 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 20 | 20 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
138 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 | C3 | C6 | C6 | C6 | D4 | D5 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C6×D5 | C3×C5⋊D4 |
| kernel | C3×C24⋊2D5 | C3×C23.D5 | C6×C5⋊D4 | C23×C30 | C24⋊2D5 | C23.D5 | C2×C5⋊D4 | C23×C10 | C2×C30 | C23×C6 | C22×C6 | C2×C10 | C24 | C2×C6 | C23 | C22 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 6 | 2 | 6 | 12 | 4 | 24 | 12 | 48 |
Matrix representation of C3×C24⋊2D5 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 47 |
| 60 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 13 | 1 |
| 1 | 0 | 0 | 0 |
| 21 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 9 | 0 | 0 | 0 |
| 12 | 34 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 10 | 34 |
| 4 | 49 | 0 | 0 |
| 47 | 57 | 0 | 0 |
| 0 | 0 | 34 | 24 |
| 0 | 0 | 51 | 27 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,47,0,0,0,0,47],[60,40,0,0,0,1,0,0,0,0,60,13,0,0,0,1],[1,21,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[9,12,0,0,0,34,0,0,0,0,9,10,0,0,0,34],[4,47,0,0,49,57,0,0,0,0,34,51,0,0,24,27] >;
C3×C24⋊2D5 in GAP, Magma, Sage, TeX
C_3\times C_2^4\rtimes_2D_5
% in TeX
G:=Group("C3xC2^4:2D5"); // GroupNames label
G:=SmallGroup(480,746);
// by ID
G=gap.SmallGroup(480,746);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,590,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=e^2=f^5=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