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## G = He3.3Dic3order 324 = 22·34

### 3rd non-split extension by He3 of Dic3 acting via Dic3/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — He3.C3 — He3.3Dic3
 Chief series C1 — C3 — C32 — C3×C9 — He3.C3 — C2×He3.C3 — He3.3Dic3
 Lower central He3.C3 — He3.3Dic3
 Upper central C1 — C2

Generators and relations for He3.3Dic3
G = < a,b,c,d,e | a3=b3=c3=1, d6=ebe-1=b-1, e2=b-1d3, ab=ba, cac-1=eae-1=ab-1, ad=da, bc=cb, bd=db, dcd-1=ab-1c, ece-1=c-1, ede-1=bd5 >

Character table of He3.3Dic3

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 9A 9B 9C 9D 9E 12A 12B 12C 12D 18A 18B 18C 18D 18E size 1 1 2 3 3 18 27 27 2 3 3 18 6 6 6 18 18 27 27 27 27 6 6 6 18 18 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 1 1 i -i -1 -1 -1 -1 1 1 1 1 1 i -i i -i -1 -1 -1 -1 -1 linear of order 4 ρ4 1 -1 1 1 1 1 -i i -1 -1 -1 -1 1 1 1 1 1 -i i -i i -1 -1 -1 -1 -1 linear of order 4 ρ5 2 2 2 2 2 -1 0 0 2 2 2 -1 -1 -1 -1 2 -1 0 0 0 0 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ6 2 2 2 2 2 2 0 0 2 2 2 2 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 2 2 -1 0 0 2 2 2 -1 -1 -1 -1 -1 2 0 0 0 0 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ8 2 2 2 2 2 -1 0 0 2 2 2 -1 2 2 2 -1 -1 0 0 0 0 2 2 2 -1 -1 orthogonal lifted from S3 ρ9 2 -2 2 2 2 -1 0 0 -2 -2 -2 1 -1 -1 -1 -1 2 0 0 0 0 1 1 1 1 -2 symplectic lifted from Dic3, Schur index 2 ρ10 2 -2 2 2 2 -1 0 0 -2 -2 -2 1 2 2 2 -1 -1 0 0 0 0 -2 -2 -2 1 1 symplectic lifted from Dic3, Schur index 2 ρ11 2 -2 2 2 2 -1 0 0 -2 -2 -2 1 -1 -1 -1 2 -1 0 0 0 0 1 1 1 -2 1 symplectic lifted from Dic3, Schur index 2 ρ12 2 -2 2 2 2 2 0 0 -2 -2 -2 -2 -1 -1 -1 -1 -1 0 0 0 0 1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ13 3 3 3 -3-3√-3/2 -3+3√-3/2 0 -1 -1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ6 ζ65 ζ65 ζ6 0 0 0 0 0 complex lifted from He3⋊C2 ρ14 3 3 3 -3+3√-3/2 -3-3√-3/2 0 -1 -1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ65 ζ6 ζ6 ζ65 0 0 0 0 0 complex lifted from He3⋊C2 ρ15 3 3 3 -3-3√-3/2 -3+3√-3/2 0 1 1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ32 ζ3 ζ3 ζ32 0 0 0 0 0 complex lifted from He3⋊C2 ρ16 3 3 3 -3+3√-3/2 -3-3√-3/2 0 1 1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ3 ζ32 ζ32 ζ3 0 0 0 0 0 complex lifted from He3⋊C2 ρ17 3 -3 3 -3-3√-3/2 -3+3√-3/2 0 -i i -3 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 0 0 0 0 0 complex lifted from He3⋊3C4 ρ18 3 -3 3 -3+3√-3/2 -3-3√-3/2 0 -i i -3 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 0 0 0 0 0 complex lifted from He3⋊3C4 ρ19 3 -3 3 -3-3√-3/2 -3+3√-3/2 0 i -i -3 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 0 0 0 0 0 complex lifted from He3⋊3C4 ρ20 3 -3 3 -3+3√-3/2 -3-3√-3/2 0 i -i -3 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 0 0 0 0 0 complex lifted from He3⋊3C4 ρ21 6 6 -3 0 0 0 0 0 -3 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 0 0 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 0 0 orthogonal lifted from He3.3S3 ρ22 6 6 -3 0 0 0 0 0 -3 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 orthogonal lifted from He3.3S3 ρ23 6 6 -3 0 0 0 0 0 -3 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 0 0 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 0 0 orthogonal lifted from He3.3S3 ρ24 6 -6 -3 0 0 0 0 0 3 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 0 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 0 0 symplectic faithful, Schur index 2 ρ25 6 -6 -3 0 0 0 0 0 3 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 0 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 symplectic faithful, Schur index 2 ρ26 6 -6 -3 0 0 0 0 0 3 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of He3.3Dic3 in GL9(𝔽37)

 26 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 26 0 0 0 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 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 1 0 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 0 0 0 0 1 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 36
,
 1 0 0 0 0 0 0 0 0 1 10 0 0 0 0 0 0 0 27 0 26 0 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 0 0 0 36 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 36
,
 12 34 0 0 0 0 0 0 0 4 25 36 0 0 0 0 0 0 2 21 0 0 0 0 0 0 0 0 0 0 14 34 20 23 14 34 0 0 0 3 17 14 34 3 17 0 0 0 14 34 14 34 20 23 0 0 0 3 17 3 17 14 34 0 0 0 20 23 14 34 14 34 0 0 0 14 34 3 17 3 17
,
 31 0 0 0 0 0 0 0 0 8 0 31 0 0 0 0 0 0 29 31 0 0 0 0 0 0 0 0 0 0 0 1 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 36 36 0 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 1

G:=sub<GL(9,GF(37))| [26,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36],[1,1,27,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36],[12,4,2,0,0,0,0,0,0,34,25,21,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,14,3,14,3,20,14,0,0,0,34,17,34,17,23,34,0,0,0,20,14,14,3,14,3,0,0,0,23,34,34,17,34,17,0,0,0,14,3,20,14,14,3,0,0,0,34,17,23,34,34,17],[31,8,29,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,1] >;

He3.3Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("He3.3Dic3");
// GroupNames label

G:=SmallGroup(324,23);
// by ID

G=gap.SmallGroup(324,23);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,146,5763,303,237,7564,1096,7781]);
// Polycyclic

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

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