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## G = (C3×C9)⋊5D9order 486 = 2·35

### 5th semidirect product of C3×C9 and D9 acting via D9/C3=S3

Aliases: (C3×C9)⋊5D9, C32⋊C9.11S3, C32.8(C9⋊S3), (C32×C9).11S3, C33.21(C3⋊S3), C3.4(C322D9), C32.19He33C2, C3.6(He3.3S3), C32.21(He3⋊C2), SmallGroup(486,53)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32.19He3 — (C3×C9)⋊5D9
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C32.19He3 — (C3×C9)⋊5D9
 Lower central C32.19He3 — (C3×C9)⋊5D9
 Upper central C1

Generators and relations for (C3×C9)⋊5D9
G = < a,b,c,d | a3=b9=c9=d2=1, ab=ba, cac-1=ab3, dad=ab6, cbc-1=ab7, dbd=ab2, dcd=c-1 >

Subgroups: 646 in 63 conjugacy classes, 17 normal (8 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C33, C3×D9, C9⋊S3, C3×C3⋊S3, C32⋊C9, C32×C9, C32⋊D9, C3×C9⋊S3, C32.19He3, (C3×C9)⋊5D9
Quotients: C1, C2, S3, D9, C3⋊S3, C9⋊S3, He3⋊C2, C322D9, He3.3S3, (C3×C9)⋊5D9

Character table of (C3×C9)⋊5D9

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L 9M 9N 9O 9P 9Q 9R size 1 81 2 2 2 2 3 3 6 6 81 81 6 6 6 6 6 6 6 6 6 18 18 18 18 18 18 18 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 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 2 0 2 2 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 orthogonal lifted from S3 ρ4 2 0 2 2 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ5 2 0 2 2 2 2 2 2 2 2 0 0 2 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 0 2 2 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 orthogonal lifted from S3 ρ7 2 0 2 -1 -1 -1 2 2 -1 -1 0 0 -1 -1 -1 2 2 2 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ8 2 0 2 -1 -1 -1 2 2 -1 -1 0 0 2 2 2 -1 -1 -1 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ9 2 0 2 -1 -1 -1 2 2 -1 -1 0 0 2 2 2 -1 -1 -1 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ10 2 0 2 -1 -1 -1 2 2 -1 -1 0 0 -1 -1 -1 -1 -1 -1 2 2 2 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ11 2 0 2 -1 -1 -1 2 2 -1 -1 0 0 -1 -1 -1 -1 -1 -1 2 2 2 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ12 2 0 2 -1 -1 -1 2 2 -1 -1 0 0 2 2 2 -1 -1 -1 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ13 2 0 2 -1 -1 -1 2 2 -1 -1 0 0 -1 -1 -1 2 2 2 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ14 2 0 2 -1 -1 -1 2 2 -1 -1 0 0 -1 -1 -1 2 2 2 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ15 2 0 2 -1 -1 -1 2 2 -1 -1 0 0 -1 -1 -1 -1 -1 -1 2 2 2 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ16 3 1 3 3 3 3 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 ζ3 ζ32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ17 3 -1 3 3 3 3 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ18 3 -1 3 3 3 3 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ19 3 1 3 3 3 3 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 ζ32 ζ3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ20 6 0 -3 6 -3 -3 0 0 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ21 6 0 -3 -3 6 -3 0 0 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ22 6 0 -3 -3 -3 6 0 0 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ23 6 0 -3 6 -3 -3 0 0 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ24 6 0 -3 -3 6 -3 0 0 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ25 6 0 -3 -3 6 -3 0 0 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ26 6 0 -3 -3 -3 6 0 0 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ27 6 0 -3 6 -3 -3 0 0 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ28 6 0 -3 -3 -3 6 0 0 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 0 0 0 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ29 6 0 6 -3 -3 -3 -3-3√-3 -3+3√-3 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊2D9 ρ30 6 0 6 -3 -3 -3 -3+3√-3 -3-3√-3 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊2D9

Smallest permutation representation of (C3×C9)⋊5D9
On 81 points
Generators in S81
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(73 79 76)(74 80 77)(75 81 78)
(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)
(1 23 68 37 78 51 29 15 63)(2 24 66 38 79 49 30 16 61)(3 25 64 39 80 47 31 17 59)(4 26 71 40 81 54 32 18 57)(5 27 69 41 73 52 33 10 55)(6 19 67 42 74 50 34 11 62)(7 20 65 43 75 48 35 12 60)(8 21 72 44 76 46 36 13 58)(9 22 70 45 77 53 28 14 56)
(1 63)(2 56)(3 58)(4 60)(5 62)(6 55)(7 57)(8 59)(9 61)(10 19)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(28 66)(29 68)(30 70)(31 72)(32 65)(33 67)(34 69)(35 71)(36 64)(37 51)(38 53)(39 46)(40 48)(41 50)(42 52)(43 54)(44 47)(45 49)(73 74)(75 81)(76 80)(77 79)

G:=sub<Sym(81)| (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(73,79,76)(74,80,77)(75,81,78), (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), (1,23,68,37,78,51,29,15,63)(2,24,66,38,79,49,30,16,61)(3,25,64,39,80,47,31,17,59)(4,26,71,40,81,54,32,18,57)(5,27,69,41,73,52,33,10,55)(6,19,67,42,74,50,34,11,62)(7,20,65,43,75,48,35,12,60)(8,21,72,44,76,46,36,13,58)(9,22,70,45,77,53,28,14,56), (1,63)(2,56)(3,58)(4,60)(5,62)(6,55)(7,57)(8,59)(9,61)(10,19)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(28,66)(29,68)(30,70)(31,72)(32,65)(33,67)(34,69)(35,71)(36,64)(37,51)(38,53)(39,46)(40,48)(41,50)(42,52)(43,54)(44,47)(45,49)(73,74)(75,81)(76,80)(77,79)>;

G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(73,79,76)(74,80,77)(75,81,78), (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), (1,23,68,37,78,51,29,15,63)(2,24,66,38,79,49,30,16,61)(3,25,64,39,80,47,31,17,59)(4,26,71,40,81,54,32,18,57)(5,27,69,41,73,52,33,10,55)(6,19,67,42,74,50,34,11,62)(7,20,65,43,75,48,35,12,60)(8,21,72,44,76,46,36,13,58)(9,22,70,45,77,53,28,14,56), (1,63)(2,56)(3,58)(4,60)(5,62)(6,55)(7,57)(8,59)(9,61)(10,19)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(28,66)(29,68)(30,70)(31,72)(32,65)(33,67)(34,69)(35,71)(36,64)(37,51)(38,53)(39,46)(40,48)(41,50)(42,52)(43,54)(44,47)(45,49)(73,74)(75,81)(76,80)(77,79) );

G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(73,79,76),(74,80,77),(75,81,78)], [(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)], [(1,23,68,37,78,51,29,15,63),(2,24,66,38,79,49,30,16,61),(3,25,64,39,80,47,31,17,59),(4,26,71,40,81,54,32,18,57),(5,27,69,41,73,52,33,10,55),(6,19,67,42,74,50,34,11,62),(7,20,65,43,75,48,35,12,60),(8,21,72,44,76,46,36,13,58),(9,22,70,45,77,53,28,14,56)], [(1,63),(2,56),(3,58),(4,60),(5,62),(6,55),(7,57),(8,59),(9,61),(10,19),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20),(28,66),(29,68),(30,70),(31,72),(32,65),(33,67),(34,69),(35,71),(36,64),(37,51),(38,53),(39,46),(40,48),(41,50),(42,52),(43,54),(44,47),(45,49),(73,74),(75,81),(76,80),(77,79)]])

Matrix representation of (C3×C9)⋊5D9 in GL8(𝔽19)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 18 0 0 0 0 0 0 0 7 10 1 0 0 0 0 0 7 10 0 1 0 0 0 0 18 8 0 0 18 18 0 0 12 18 0 0 1 0
,
 0 18 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 0 5 12 0 0 0 0 0 0 7 17 0 0 0 0 0 0 6 5 7 5 0 0 0 0 2 3 14 2 0 0 0 0 11 7 0 0 17 12 0 0 7 1 0 0 7 5
,
 14 17 0 0 0 0 0 0 2 12 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 3 11 17 18 0 0 0 0 9 12 17 10 1 0 0 0 9 12 17 10 0 1 0 0 16 1 11 18 0 0 0 0 15 2 11 18 0 0
,
 7 14 0 0 0 0 0 0 2 12 0 0 0 0 0 0 0 0 3 11 17 18 0 0 0 0 0 0 18 1 0 0 0 0 9 17 7 9 0 0 0 0 9 18 7 9 0 0 0 0 10 9 12 1 18 18 0 0 4 0 12 1 0 1

G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,18,7,7,18,12,0,0,1,0,10,10,8,18,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0],[0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,5,7,6,2,11,7,0,0,12,17,5,3,7,1,0,0,0,0,7,14,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,12,5],[14,2,0,0,0,0,0,0,17,12,0,0,0,0,0,0,0,0,0,3,9,9,16,15,0,0,0,11,12,12,1,2,0,0,18,17,17,17,11,11,0,0,1,18,10,10,18,18,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[7,2,0,0,0,0,0,0,14,12,0,0,0,0,0,0,0,0,3,0,9,9,10,4,0,0,11,0,17,18,9,0,0,0,17,18,7,7,12,12,0,0,18,1,9,9,1,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,1] >;

(C3×C9)⋊5D9 in GAP, Magma, Sage, TeX

(C_3\times C_9)\rtimes_5D_9
% in TeX

G:=Group("(C3xC9):5D9");
// GroupNames label

G:=SmallGroup(486,53);
// by ID

G=gap.SmallGroup(486,53);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,265,223,6050,548,500,867,11344,3250,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=c^9=d^2=1,a*b=b*a,c*a*c^-1=a*b^3,d*a*d=a*b^6,c*b*c^-1=a*b^7,d*b*d=a*b^2,d*c*d=c^-1>;
// generators/relations

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