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## G = C9⋊Dic3order 108 = 22·33

### The semidirect product of C9 and Dic3 acting via Dic3/C6=C2

Aliases: C9⋊Dic3, C3⋊Dic9, C6.3D9, C18.3S3, C32.3Dic3, (C3×C9)⋊3C4, C2.(C9⋊S3), (C3×C6).6S3, C6.1(C3⋊S3), (C3×C18).3C2, C3.(C3⋊Dic3), SmallGroup(108,10)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C9⋊Dic3
G = < a,b,c | a9=b6=1, c2=b3, ab=ba, cac-1=a-1, cbc-1=b-1 >

Character table of C9⋊Dic3

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 9A 9B 9C 9D 9E 9F 9G 9H 9I 18A 18B 18C 18D 18E 18F 18G 18H 18I size 1 1 2 2 2 2 27 27 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 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 1 1 1 1 -1 -1 -1 -1 -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 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ5 2 2 -1 -1 -1 2 0 0 -1 2 -1 -1 2 -1 -1 2 2 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 2 2 2 0 0 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 -1 -1 -1 2 0 0 -1 2 -1 -1 -1 2 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 orthogonal lifted from S3 ρ8 2 2 2 -1 -1 -1 0 0 -1 -1 2 -1 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ9 2 2 -1 -1 -1 2 0 0 -1 2 -1 -1 -1 -1 2 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 2 2 orthogonal lifted from S3 ρ10 2 2 -1 -1 2 -1 0 0 2 -1 -1 -1 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ11 2 2 2 -1 -1 -1 0 0 -1 -1 2 -1 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ12 2 2 2 -1 -1 -1 0 0 -1 -1 2 -1 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ13 2 2 -1 -1 2 -1 0 0 2 -1 -1 -1 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ14 2 2 -1 2 -1 -1 0 0 -1 -1 -1 2 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ15 2 2 -1 2 -1 -1 0 0 -1 -1 -1 2 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ16 2 2 -1 -1 2 -1 0 0 2 -1 -1 -1 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ17 2 2 -1 2 -1 -1 0 0 -1 -1 -1 2 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ18 2 -2 -1 -1 -1 2 0 0 1 -2 1 1 2 -1 -1 2 2 -1 -1 -1 -1 1 -2 -2 -2 1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ19 2 -2 -1 -1 -1 2 0 0 1 -2 1 1 -1 2 -1 -1 -1 2 2 -1 -1 1 1 1 1 -2 -2 -2 1 1 symplectic lifted from Dic3, Schur index 2 ρ20 2 -2 2 -1 -1 -1 0 0 1 1 -2 1 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 symplectic lifted from Dic9, Schur index 2 ρ21 2 -2 -1 -1 -1 2 0 0 1 -2 1 1 -1 -1 2 -1 -1 -1 -1 2 2 -2 1 1 1 1 1 1 -2 -2 symplectic lifted from Dic3, Schur index 2 ρ22 2 -2 -1 2 -1 -1 0 0 1 1 1 -2 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 symplectic lifted from Dic9, Schur index 2 ρ23 2 -2 -1 2 -1 -1 0 0 1 1 1 -2 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 symplectic lifted from Dic9, Schur index 2 ρ24 2 -2 -1 -1 2 -1 0 0 -2 1 1 1 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 symplectic lifted from Dic9, Schur index 2 ρ25 2 -2 -1 -1 2 -1 0 0 -2 1 1 1 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 symplectic lifted from Dic9, Schur index 2 ρ26 2 -2 2 -1 -1 -1 0 0 1 1 -2 1 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 symplectic lifted from Dic9, Schur index 2 ρ27 2 -2 2 2 2 2 0 0 -2 -2 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ28 2 -2 -1 2 -1 -1 0 0 1 1 1 -2 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 symplectic lifted from Dic9, Schur index 2 ρ29 2 -2 -1 -1 2 -1 0 0 -2 1 1 1 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 symplectic lifted from Dic9, Schur index 2 ρ30 2 -2 2 -1 -1 -1 0 0 1 1 -2 1 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 symplectic lifted from Dic9, Schur index 2

Smallest permutation representation of C9⋊Dic3
Regular action on 108 points
Generators in S108
(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 48 35 59 37 65)(2 49 36 60 38 66)(3 50 28 61 39 67)(4 51 29 62 40 68)(5 52 30 63 41 69)(6 53 31 55 42 70)(7 54 32 56 43 71)(8 46 33 57 44 72)(9 47 34 58 45 64)(10 80 25 98 107 84)(11 81 26 99 108 85)(12 73 27 91 100 86)(13 74 19 92 101 87)(14 75 20 93 102 88)(15 76 21 94 103 89)(16 77 22 95 104 90)(17 78 23 96 105 82)(18 79 24 97 106 83)
(1 23 59 82)(2 22 60 90)(3 21 61 89)(4 20 62 88)(5 19 63 87)(6 27 55 86)(7 26 56 85)(8 25 57 84)(9 24 58 83)(10 72 98 33)(11 71 99 32)(12 70 91 31)(13 69 92 30)(14 68 93 29)(15 67 94 28)(16 66 95 36)(17 65 96 35)(18 64 97 34)(37 105 48 78)(38 104 49 77)(39 103 50 76)(40 102 51 75)(41 101 52 74)(42 100 53 73)(43 108 54 81)(44 107 46 80)(45 106 47 79)

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

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

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

C9⋊Dic3 is a maximal subgroup of
C9⋊Dic6  Dic3×D9  S3×Dic9  D6⋊D9  C12.D9  C4×C9⋊S3  C6.D18  C32⋊Dic9  He3.Dic3  He3.2Dic3  C9⋊Dic9  C27⋊Dic3  C33.Dic3  He3.4Dic3  C325Dic9  C18.5S4  A4⋊Dic9  C32.3CSU2(𝔽3)  C62.10Dic3
C9⋊Dic3 is a maximal quotient of
C36.S3  C9⋊Dic9  C322Dic9  C27⋊Dic3  C325Dic9  A4⋊Dic9  C62.10Dic3

Matrix representation of C9⋊Dic3 in GL4(𝔽37) generated by

 17 11 0 0 26 6 0 0 0 0 26 6 0 0 31 20
,
 1 1 0 0 36 0 0 0 0 0 0 1 0 0 36 36
,
 7 14 0 0 7 30 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(37))| [17,26,0,0,11,6,0,0,0,0,26,31,0,0,6,20],[1,36,0,0,1,0,0,0,0,0,0,36,0,0,1,36],[7,7,0,0,14,30,0,0,0,0,0,1,0,0,1,0] >;

C9⋊Dic3 in GAP, Magma, Sage, TeX

C_9\rtimes {\rm Dic}_3
% in TeX

G:=Group("C9:Dic3");
// GroupNames label

G:=SmallGroup(108,10);
// by ID

G=gap.SmallGroup(108,10);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,10,662,282,483,1804]);
// Polycyclic

G:=Group<a,b,c|a^9=b^6=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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