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## G = C5⋊U2(𝔽3)  order 480 = 25·3·5

### The semidirect product of C5 and U2(𝔽3) acting via U2(𝔽3)/SL2(𝔽3)=C4

Aliases: C5⋊U2(𝔽3), Dic5.4S4, SL2(𝔽3)⋊1F5, Q8.(C3⋊F5), (C5×Q8).Dic3, C2.2(A4⋊F5), C10.1(A4⋊C4), Q82D5.2S3, Dic5.A4.2C2, (C5×SL2(𝔽3))⋊1C4, SmallGroup(480,961)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — C5⋊U2(𝔽3)
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Dic5.A4 — C5⋊U2(𝔽3)
 Lower central C5×SL2(𝔽3) — C5⋊U2(𝔽3)
 Upper central C1 — C2

Generators and relations for C5⋊U2(𝔽3)
G = < a,b,c,d,e,f | a5=b4=e3=1, c2=d2=b2, f2=b, bab-1=a-1, ac=ca, ad=da, ae=ea, faf-1=a2, bc=cb, bd=db, be=eb, bf=fb, dcd-1=b2c, ece-1=b2cd, fcf-1=cd, ede-1=c, fdf-1=b2d, fef-1=e-1 >

30C2
4C3
3C4
5C4
15C22
30C4
30C4
4C6
6D5
4C15
15C2×C4
15D4
30C8
30C2×C4
20C12
3D10
3C20
6F5
6F5
4C30
15M4(2)
15C42
20C3⋊C8
3D20
15C4≀C2

Character table of C5⋊U2(𝔽3)

 class 1 2A 2B 3 4A 4B 4C 4D 4E 4F 4G 5 6 8A 8B 10 12A 12B 15A 15B 20 30A 30B size 1 1 30 8 5 5 6 30 30 30 30 4 8 60 60 4 40 40 16 16 24 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 -1 -1 1 -i i -i i 1 1 -i i 1 -1 -1 1 1 1 1 1 linear of order 4 ρ4 1 1 -1 1 -1 -1 1 i -i i -i 1 1 i -i 1 -1 -1 1 1 1 1 1 linear of order 4 ρ5 2 2 2 -1 2 2 2 0 0 0 0 2 -1 0 0 2 -1 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ6 2 2 -2 -1 -2 -2 2 0 0 0 0 2 -1 0 0 2 1 1 -1 -1 2 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ7 2 -2 0 -1 -2i 2i 0 1-i 1+i -1+i -1-i 2 1 0 0 -2 -i i -1 -1 0 1 1 complex lifted from U2(𝔽3) ρ8 2 -2 0 -1 2i -2i 0 -1-i -1+i 1+i 1-i 2 1 0 0 -2 i -i -1 -1 0 1 1 complex lifted from U2(𝔽3) ρ9 2 -2 0 -1 2i -2i 0 1+i 1-i -1-i -1+i 2 1 0 0 -2 i -i -1 -1 0 1 1 complex lifted from U2(𝔽3) ρ10 2 -2 0 -1 -2i 2i 0 -1+i -1-i 1-i 1+i 2 1 0 0 -2 -i i -1 -1 0 1 1 complex lifted from U2(𝔽3) ρ11 3 3 -1 0 3 3 -1 1 1 1 1 3 0 -1 -1 3 0 0 0 0 -1 0 0 orthogonal lifted from S4 ρ12 3 3 -1 0 3 3 -1 -1 -1 -1 -1 3 0 1 1 3 0 0 0 0 -1 0 0 orthogonal lifted from S4 ρ13 3 3 1 0 -3 -3 -1 -i i -i i 3 0 i -i 3 0 0 0 0 -1 0 0 complex lifted from A4⋊C4 ρ14 3 3 1 0 -3 -3 -1 i -i i -i 3 0 -i i 3 0 0 0 0 -1 0 0 complex lifted from A4⋊C4 ρ15 4 4 0 4 0 0 4 0 0 0 0 -1 4 0 0 -1 0 0 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ16 4 -4 0 1 -4i 4i 0 0 0 0 0 4 -1 0 0 -4 i -i 1 1 0 -1 -1 complex lifted from U2(𝔽3) ρ17 4 -4 0 1 4i -4i 0 0 0 0 0 4 -1 0 0 -4 -i i 1 1 0 -1 -1 complex lifted from U2(𝔽3) ρ18 4 4 0 -2 0 0 4 0 0 0 0 -1 -2 0 0 -1 0 0 1-√-15/2 1+√-15/2 -1 1+√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ19 4 4 0 -2 0 0 4 0 0 0 0 -1 -2 0 0 -1 0 0 1+√-15/2 1-√-15/2 -1 1-√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ20 8 -8 0 -4 0 0 0 0 0 0 0 -2 4 0 0 2 0 0 1 1 0 -1 -1 orthogonal faithful ρ21 8 -8 0 2 0 0 0 0 0 0 0 -2 -2 0 0 2 0 0 -1-√-15/2 -1+√-15/2 0 1-√-15/2 1+√-15/2 complex faithful ρ22 8 -8 0 2 0 0 0 0 0 0 0 -2 -2 0 0 2 0 0 -1+√-15/2 -1-√-15/2 0 1+√-15/2 1-√-15/2 complex faithful ρ23 12 12 0 0 0 0 -4 0 0 0 0 -3 0 0 0 -3 0 0 0 0 1 0 0 orthogonal lifted from A4⋊F5

Smallest permutation representation of C5⋊U2(𝔽3)
On 120 points
Generators in S120
(1 16 28 18 38)(2 29 39 9 19)(3 40 20 30 10)(4 21 11 33 31)(5 12 32 22 34)(6 25 35 13 23)(7 36 24 26 14)(8 17 15 37 27)(41 51 85 73 103)(42 86 104 52 74)(43 97 75 87 53)(44 76 54 98 88)(45 55 81 77 99)(46 82 100 56 78)(47 101 79 83 49)(48 80 50 102 84)(57 95 117 67 109)(58 118 110 96 68)(59 111 69 119 89)(60 70 90 112 120)(61 91 113 71 105)(62 114 106 92 72)(63 107 65 115 93)(64 66 94 108 116)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)
(1 4 5 8)(2 3 6 7)(9 30 13 26)(10 23 14 19)(11 32 15 28)(12 17 16 21)(18 33 22 37)(20 35 24 39)(25 36 29 40)(27 38 31 34)(41 108 45 112)(42 44 46 48)(43 110 47 106)(49 114 53 118)(50 104 54 100)(51 116 55 120)(52 98 56 102)(57 65 61 69)(58 83 62 87)(59 67 63 71)(60 85 64 81)(66 77 70 73)(68 79 72 75)(74 88 78 84)(76 82 80 86)(89 117 93 113)(90 103 94 99)(91 119 95 115)(92 97 96 101)(105 111 109 107)
(1 3 5 7)(2 8 6 4)(9 37 13 33)(10 34 14 38)(11 39 15 35)(12 36 16 40)(17 25 21 29)(18 30 22 26)(19 27 23 31)(20 32 24 28)(41 110 45 106)(42 107 46 111)(43 112 47 108)(44 109 48 105)(49 94 53 90)(50 91 54 95)(51 96 55 92)(52 93 56 89)(57 80 61 76)(58 77 62 73)(59 74 63 78)(60 79 64 75)(65 82 69 86)(66 87 70 83)(67 84 71 88)(68 81 72 85)(97 120 101 116)(98 117 102 113)(99 114 103 118)(100 119 104 115)
(1 110 42)(2 43 111)(3 112 44)(4 45 105)(5 106 46)(6 47 107)(7 108 48)(8 41 109)(9 87 89)(10 90 88)(11 81 91)(12 92 82)(13 83 93)(14 94 84)(15 85 95)(16 96 86)(17 51 57)(18 58 52)(19 53 59)(20 60 54)(21 55 61)(22 62 56)(23 49 63)(24 64 50)(25 101 65)(26 66 102)(27 103 67)(28 68 104)(29 97 69)(30 70 98)(31 99 71)(32 72 100)(33 77 113)(34 114 78)(35 79 115)(36 116 80)(37 73 117)(38 118 74)(39 75 119)(40 120 76)
(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)

G:=sub<Sym(120)| (1,16,28,18,38)(2,29,39,9,19)(3,40,20,30,10)(4,21,11,33,31)(5,12,32,22,34)(6,25,35,13,23)(7,36,24,26,14)(8,17,15,37,27)(41,51,85,73,103)(42,86,104,52,74)(43,97,75,87,53)(44,76,54,98,88)(45,55,81,77,99)(46,82,100,56,78)(47,101,79,83,49)(48,80,50,102,84)(57,95,117,67,109)(58,118,110,96,68)(59,111,69,119,89)(60,70,90,112,120)(61,91,113,71,105)(62,114,106,92,72)(63,107,65,115,93)(64,66,94,108,116), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120), (1,4,5,8)(2,3,6,7)(9,30,13,26)(10,23,14,19)(11,32,15,28)(12,17,16,21)(18,33,22,37)(20,35,24,39)(25,36,29,40)(27,38,31,34)(41,108,45,112)(42,44,46,48)(43,110,47,106)(49,114,53,118)(50,104,54,100)(51,116,55,120)(52,98,56,102)(57,65,61,69)(58,83,62,87)(59,67,63,71)(60,85,64,81)(66,77,70,73)(68,79,72,75)(74,88,78,84)(76,82,80,86)(89,117,93,113)(90,103,94,99)(91,119,95,115)(92,97,96,101)(105,111,109,107), (1,3,5,7)(2,8,6,4)(9,37,13,33)(10,34,14,38)(11,39,15,35)(12,36,16,40)(17,25,21,29)(18,30,22,26)(19,27,23,31)(20,32,24,28)(41,110,45,106)(42,107,46,111)(43,112,47,108)(44,109,48,105)(49,94,53,90)(50,91,54,95)(51,96,55,92)(52,93,56,89)(57,80,61,76)(58,77,62,73)(59,74,63,78)(60,79,64,75)(65,82,69,86)(66,87,70,83)(67,84,71,88)(68,81,72,85)(97,120,101,116)(98,117,102,113)(99,114,103,118)(100,119,104,115), (1,110,42)(2,43,111)(3,112,44)(4,45,105)(5,106,46)(6,47,107)(7,108,48)(8,41,109)(9,87,89)(10,90,88)(11,81,91)(12,92,82)(13,83,93)(14,94,84)(15,85,95)(16,96,86)(17,51,57)(18,58,52)(19,53,59)(20,60,54)(21,55,61)(22,62,56)(23,49,63)(24,64,50)(25,101,65)(26,66,102)(27,103,67)(28,68,104)(29,97,69)(30,70,98)(31,99,71)(32,72,100)(33,77,113)(34,114,78)(35,79,115)(36,116,80)(37,73,117)(38,118,74)(39,75,119)(40,120,76), (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)>;

G:=Group( (1,16,28,18,38)(2,29,39,9,19)(3,40,20,30,10)(4,21,11,33,31)(5,12,32,22,34)(6,25,35,13,23)(7,36,24,26,14)(8,17,15,37,27)(41,51,85,73,103)(42,86,104,52,74)(43,97,75,87,53)(44,76,54,98,88)(45,55,81,77,99)(46,82,100,56,78)(47,101,79,83,49)(48,80,50,102,84)(57,95,117,67,109)(58,118,110,96,68)(59,111,69,119,89)(60,70,90,112,120)(61,91,113,71,105)(62,114,106,92,72)(63,107,65,115,93)(64,66,94,108,116), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120), (1,4,5,8)(2,3,6,7)(9,30,13,26)(10,23,14,19)(11,32,15,28)(12,17,16,21)(18,33,22,37)(20,35,24,39)(25,36,29,40)(27,38,31,34)(41,108,45,112)(42,44,46,48)(43,110,47,106)(49,114,53,118)(50,104,54,100)(51,116,55,120)(52,98,56,102)(57,65,61,69)(58,83,62,87)(59,67,63,71)(60,85,64,81)(66,77,70,73)(68,79,72,75)(74,88,78,84)(76,82,80,86)(89,117,93,113)(90,103,94,99)(91,119,95,115)(92,97,96,101)(105,111,109,107), (1,3,5,7)(2,8,6,4)(9,37,13,33)(10,34,14,38)(11,39,15,35)(12,36,16,40)(17,25,21,29)(18,30,22,26)(19,27,23,31)(20,32,24,28)(41,110,45,106)(42,107,46,111)(43,112,47,108)(44,109,48,105)(49,94,53,90)(50,91,54,95)(51,96,55,92)(52,93,56,89)(57,80,61,76)(58,77,62,73)(59,74,63,78)(60,79,64,75)(65,82,69,86)(66,87,70,83)(67,84,71,88)(68,81,72,85)(97,120,101,116)(98,117,102,113)(99,114,103,118)(100,119,104,115), (1,110,42)(2,43,111)(3,112,44)(4,45,105)(5,106,46)(6,47,107)(7,108,48)(8,41,109)(9,87,89)(10,90,88)(11,81,91)(12,92,82)(13,83,93)(14,94,84)(15,85,95)(16,96,86)(17,51,57)(18,58,52)(19,53,59)(20,60,54)(21,55,61)(22,62,56)(23,49,63)(24,64,50)(25,101,65)(26,66,102)(27,103,67)(28,68,104)(29,97,69)(30,70,98)(31,99,71)(32,72,100)(33,77,113)(34,114,78)(35,79,115)(36,116,80)(37,73,117)(38,118,74)(39,75,119)(40,120,76), (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) );

G=PermutationGroup([[(1,16,28,18,38),(2,29,39,9,19),(3,40,20,30,10),(4,21,11,33,31),(5,12,32,22,34),(6,25,35,13,23),(7,36,24,26,14),(8,17,15,37,27),(41,51,85,73,103),(42,86,104,52,74),(43,97,75,87,53),(44,76,54,98,88),(45,55,81,77,99),(46,82,100,56,78),(47,101,79,83,49),(48,80,50,102,84),(57,95,117,67,109),(58,118,110,96,68),(59,111,69,119,89),(60,70,90,112,120),(61,91,113,71,105),(62,114,106,92,72),(63,107,65,115,93),(64,66,94,108,116)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120)], [(1,4,5,8),(2,3,6,7),(9,30,13,26),(10,23,14,19),(11,32,15,28),(12,17,16,21),(18,33,22,37),(20,35,24,39),(25,36,29,40),(27,38,31,34),(41,108,45,112),(42,44,46,48),(43,110,47,106),(49,114,53,118),(50,104,54,100),(51,116,55,120),(52,98,56,102),(57,65,61,69),(58,83,62,87),(59,67,63,71),(60,85,64,81),(66,77,70,73),(68,79,72,75),(74,88,78,84),(76,82,80,86),(89,117,93,113),(90,103,94,99),(91,119,95,115),(92,97,96,101),(105,111,109,107)], [(1,3,5,7),(2,8,6,4),(9,37,13,33),(10,34,14,38),(11,39,15,35),(12,36,16,40),(17,25,21,29),(18,30,22,26),(19,27,23,31),(20,32,24,28),(41,110,45,106),(42,107,46,111),(43,112,47,108),(44,109,48,105),(49,94,53,90),(50,91,54,95),(51,96,55,92),(52,93,56,89),(57,80,61,76),(58,77,62,73),(59,74,63,78),(60,79,64,75),(65,82,69,86),(66,87,70,83),(67,84,71,88),(68,81,72,85),(97,120,101,116),(98,117,102,113),(99,114,103,118),(100,119,104,115)], [(1,110,42),(2,43,111),(3,112,44),(4,45,105),(5,106,46),(6,47,107),(7,108,48),(8,41,109),(9,87,89),(10,90,88),(11,81,91),(12,92,82),(13,83,93),(14,94,84),(15,85,95),(16,96,86),(17,51,57),(18,58,52),(19,53,59),(20,60,54),(21,55,61),(22,62,56),(23,49,63),(24,64,50),(25,101,65),(26,66,102),(27,103,67),(28,68,104),(29,97,69),(30,70,98),(31,99,71),(32,72,100),(33,77,113),(34,114,78),(35,79,115),(36,116,80),(37,73,117),(38,118,74),(39,75,119),(40,120,76)], [(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)]])

Matrix representation of C5⋊U2(𝔽3) in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 240 240 240
,
 64 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 240 240 240 240 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 64 0 0 0 0 0 0 177 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 88 152 0 0 0 0 153 152 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 64 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 240 240 240 240 0 0 0 1 0 0

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,240,0,1,0,0,0,240,1,0],[0,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[88,153,0,0,0,0,152,152,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,64,0,0,0,0,1,0,0,0,0,0,0,0,1,0,240,0,0,0,0,0,240,1,0,0,0,1,240,0,0,0,0,0,240,0] >;

C5⋊U2(𝔽3) in GAP, Magma, Sage, TeX

C_5\rtimes {\rm U}_2({\mathbb F}_3)
% in TeX

G:=Group("C5:U(2,3)");
// GroupNames label

G:=SmallGroup(480,961);
// by ID

G=gap.SmallGroup(480,961);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,14,1688,170,1011,682,4204,3168,172,2525,1909,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^4=e^3=1,c^2=d^2=b^2,f^2=b,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a^2,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=b^2*d,f*e*f^-1=e^-1>;
// generators/relations

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