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## G = He3.2Dic3order 324 = 22·34

### 2nd non-split extension by He3 of Dic3 acting via Dic3/C2=S3

Aliases: He3.2Dic3, (C3×C9)⋊3C12, (C3×C18).3C6, C9⋊Dic33C3, He3⋊C32C4, (C2×He3).3S3, C6.4(C32⋊C6), C3.4(C32⋊C12), C2.(He3.2S3), C32.8(C3×Dic3), (C3×C6).15(C3×S3), (C2×He3⋊C3).2C2, SmallGroup(324,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — He3.2Dic3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C2×He3⋊C3 — He3.2Dic3
 Lower central C3×C9 — He3.2Dic3
 Upper central C1 — C2

Generators and relations for He3.2Dic3
G = < a,b,c,d,e | a3=b3=c3=1, d6=b, e2=bd3, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1, bc=cb, bd=db, ebe-1=b-1, dcd-1=ab-1c, ce=ec, ede-1=b-1d5 >

Character table of He3.2Dic3

 class 1 2 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 9A 9B 9C 12A 12B 12C 12D 18A 18B 18C size 1 1 2 6 9 9 18 18 27 27 2 6 9 9 18 18 6 6 6 27 27 27 27 6 6 6 ρ1 1 1 1 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 1 1 1 linear of order 2 ρ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 -1 -1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 ζ65 ζ6 ζ6 ζ65 1 1 1 linear of order 6 ρ4 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 linear of order 3 ρ5 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 linear of order 3 ρ6 1 1 1 1 ζ3 ζ32 ζ3 ζ32 -1 -1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 ζ6 ζ65 ζ65 ζ6 1 1 1 linear of order 6 ρ7 1 -1 1 1 1 1 1 1 -i i -1 -1 -1 -1 -1 -1 1 1 1 -i i -i i -1 -1 -1 linear of order 4 ρ8 1 -1 1 1 1 1 1 1 i -i -1 -1 -1 -1 -1 -1 1 1 1 i -i i -i -1 -1 -1 linear of order 4 ρ9 1 -1 1 1 ζ3 ζ32 ζ3 ζ32 -i i -1 -1 ζ65 ζ6 ζ65 ζ6 1 1 1 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 -1 -1 -1 linear of order 12 ρ10 1 -1 1 1 ζ32 ζ3 ζ32 ζ3 i -i -1 -1 ζ6 ζ65 ζ6 ζ65 1 1 1 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 -1 -1 -1 linear of order 12 ρ11 1 -1 1 1 ζ3 ζ32 ζ3 ζ32 i -i -1 -1 ζ65 ζ6 ζ65 ζ6 1 1 1 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 -1 -1 -1 linear of order 12 ρ12 1 -1 1 1 ζ32 ζ3 ζ32 ζ3 -i i -1 -1 ζ6 ζ65 ζ6 ζ65 1 1 1 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 -1 -1 -1 linear of order 12 ρ13 2 2 2 2 2 2 -1 -1 0 0 2 2 2 2 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ14 2 -2 2 2 2 2 -1 -1 0 0 -2 -2 -2 -2 1 1 -1 -1 -1 0 0 0 0 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 2 2 -1+√-3 -1-√-3 ζ65 ζ6 0 0 2 2 -1+√-3 -1-√-3 ζ65 ζ6 -1 -1 -1 0 0 0 0 -1 -1 -1 complex lifted from C3×S3 ρ16 2 -2 2 2 -1-√-3 -1+√-3 ζ6 ζ65 0 0 -2 -2 1+√-3 1-√-3 ζ32 ζ3 -1 -1 -1 0 0 0 0 1 1 1 complex lifted from C3×Dic3 ρ17 2 2 2 2 -1-√-3 -1+√-3 ζ6 ζ65 0 0 2 2 -1-√-3 -1+√-3 ζ6 ζ65 -1 -1 -1 0 0 0 0 -1 -1 -1 complex lifted from C3×S3 ρ18 2 -2 2 2 -1+√-3 -1-√-3 ζ65 ζ6 0 0 -2 -2 1-√-3 1+√-3 ζ3 ζ32 -1 -1 -1 0 0 0 0 1 1 1 complex lifted from C3×Dic3 ρ19 6 6 6 -3 0 0 0 0 0 0 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ20 6 6 -3 0 0 0 0 0 0 0 -3 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 orthogonal lifted from He3.2S3 ρ21 6 6 -3 0 0 0 0 0 0 0 -3 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 orthogonal lifted from He3.2S3 ρ22 6 6 -3 0 0 0 0 0 0 0 -3 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 orthogonal lifted from He3.2S3 ρ23 6 -6 6 -3 0 0 0 0 0 0 -6 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊C12, Schur index 2 ρ24 6 -6 -3 0 0 0 0 0 0 0 3 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 symplectic faithful, Schur index 2 ρ25 6 -6 -3 0 0 0 0 0 0 0 3 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 symplectic faithful, Schur index 2 ρ26 6 -6 -3 0 0 0 0 0 0 0 3 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 symplectic faithful, Schur index 2

Smallest permutation representation of He3.2Dic3
On 108 points
Generators in S108
(1 29 78)(2 30 79)(3 31 80)(4 32 81)(5 33 82)(6 34 83)(7 35 84)(8 36 85)(9 19 86)(10 20 87)(11 21 88)(12 22 89)(13 23 90)(14 24 73)(15 25 74)(16 26 75)(17 27 76)(18 28 77)(37 93 67)(38 94 68)(39 95 69)(40 96 70)(41 97 71)(42 98 72)(43 99 55)(44 100 56)(45 101 57)(46 102 58)(47 103 59)(48 104 60)(49 105 61)(50 106 62)(51 107 63)(52 108 64)(53 91 65)(54 92 66)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 43 49)(38 44 50)(39 45 51)(40 46 52)(41 47 53)(42 48 54)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)(73 79 85)(74 80 86)(75 81 87)(76 82 88)(77 83 89)(78 84 90)(91 97 103)(92 98 104)(93 99 105)(94 100 106)(95 101 107)(96 102 108)
(1 9 76)(2 20 34)(3 88 13)(4 12 79)(5 23 19)(6 73 16)(7 15 82)(8 26 22)(10 18 85)(11 29 25)(14 32 28)(17 35 31)(21 84 86)(24 87 89)(27 90 74)(30 75 77)(33 78 80)(36 81 83)(37 71 69)(38 42 102)(39 99 53)(40 56 72)(41 45 105)(43 59 57)(44 48 108)(46 62 60)(47 51 93)(49 65 63)(50 54 96)(52 68 66)(55 97 107)(58 100 92)(61 103 95)(64 106 98)(67 91 101)(70 94 104)
(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)
(1 97 10 106)(2 96 11 105)(3 95 12 104)(4 94 13 103)(5 93 14 102)(6 92 15 101)(7 91 16 100)(8 108 17 99)(9 107 18 98)(19 51 28 42)(20 50 29 41)(21 49 30 40)(22 48 31 39)(23 47 32 38)(24 46 33 37)(25 45 34 54)(26 44 35 53)(27 43 36 52)(55 85 64 76)(56 84 65 75)(57 83 66 74)(58 82 67 73)(59 81 68 90)(60 80 69 89)(61 79 70 88)(62 78 71 87)(63 77 72 86)

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

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

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

Matrix representation of He3.2Dic3 in GL8(𝔽37)

 1 0 0 0 0 0 0 0 0 1 0 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 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 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 36 36 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 17 14 34 20 34 20 0 0 23 3 17 14 17 14 0 0 23 3 34 20 23 3 0 0 34 20 17 14 34 20 0 0 17 14 17 14 23 3 0 0 23 3 23 3 34 20
,
 11 0 0 0 0 0 0 0 23 27 0 0 0 0 0 0 0 0 3 23 20 34 3 23 0 0 14 17 3 23 14 17 0 0 3 23 3 23 20 34 0 0 14 17 14 17 3 23 0 0 20 34 3 23 3 23 0 0 3 23 14 17 14 17
,
 2 3 0 0 0 0 0 0 23 35 0 0 0 0 0 0 0 0 16 29 34 35 32 22 0 0 13 21 1 3 27 5 0 0 34 35 32 22 16 29 0 0 1 3 27 5 13 21 0 0 32 22 16 29 34 35 0 0 27 5 13 21 1 3

G:=sub<GL(8,GF(37))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,17,23,23,34,17,23,0,0,14,3,3,20,14,3,0,0,34,17,34,17,17,23,0,0,20,14,20,14,14,3,0,0,34,17,23,34,23,34,0,0,20,14,3,20,3,20],[11,23,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,3,14,3,14,20,3,0,0,23,17,23,17,34,23,0,0,20,3,3,14,3,14,0,0,34,23,23,17,23,17,0,0,3,14,20,3,3,14,0,0,23,17,34,23,23,17],[2,23,0,0,0,0,0,0,3,35,0,0,0,0,0,0,0,0,16,13,34,1,32,27,0,0,29,21,35,3,22,5,0,0,34,1,32,27,16,13,0,0,35,3,22,5,29,21,0,0,32,27,16,13,34,1,0,0,22,5,29,21,35,3] >;

He3.2Dic3 in GAP, Magma, Sage, TeX

{\rm He}_3._2{\rm Dic}_3
% in TeX

G:=Group("He3.2Dic3");
// GroupNames label

G:=SmallGroup(324,18);
// by ID

G=gap.SmallGroup(324,18);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,3171,585,453,2164,2170,7781]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=1,d^6=b,e^2=b*d^3,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e^-1=a^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,d*c*d^-1=a*b^-1*c,c*e=e*c,e*d*e^-1=b^-1*d^5>;
// generators/relations

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