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## G = C18×C3⋊S3order 324 = 22·34

### Direct product of C18 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C18×C3⋊S3
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C9×C3⋊S3 — C18×C3⋊S3
 Lower central C32 — C18×C3⋊S3
 Upper central C1 — C18

Generators and relations for C18×C3⋊S3
G = < a,b,c,d | a18=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 298 in 122 conjugacy classes, 45 normal (15 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C9, C9, C32, C32, C32, D6, C2×C6, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C33, C2×C18, S3×C6, C2×C3⋊S3, S3×C9, C3×C18, C3×C18, C3×C3⋊S3, C32×C6, C32×C9, S3×C18, C6×C3⋊S3, C9×C3⋊S3, C32×C18, C18×C3⋊S3
Quotients: C1, C2, C3, C22, S3, C6, C9, D6, C2×C6, C18, C3×S3, C3⋊S3, C2×C18, S3×C6, C2×C3⋊S3, S3×C9, C3×C3⋊S3, S3×C18, C6×C3⋊S3, C9×C3⋊S3, C18×C3⋊S3

Smallest permutation representation of C18×C3⋊S3
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 53 73)(2 54 74)(3 37 75)(4 38 76)(5 39 77)(6 40 78)(7 41 79)(8 42 80)(9 43 81)(10 44 82)(11 45 83)(12 46 84)(13 47 85)(14 48 86)(15 49 87)(16 50 88)(17 51 89)(18 52 90)(19 68 96)(20 69 97)(21 70 98)(22 71 99)(23 72 100)(24 55 101)(25 56 102)(26 57 103)(27 58 104)(28 59 105)(29 60 106)(30 61 107)(31 62 108)(32 63 91)(33 64 92)(34 65 93)(35 66 94)(36 67 95)
(1 41 85)(2 42 86)(3 43 87)(4 44 88)(5 45 89)(6 46 90)(7 47 73)(8 48 74)(9 49 75)(10 50 76)(11 51 77)(12 52 78)(13 53 79)(14 54 80)(15 37 81)(16 38 82)(17 39 83)(18 40 84)(19 62 102)(20 63 103)(21 64 104)(22 65 105)(23 66 106)(24 67 107)(25 68 108)(26 69 91)(27 70 92)(28 71 93)(29 72 94)(30 55 95)(31 56 96)(32 57 97)(33 58 98)(34 59 99)(35 60 100)(36 61 101)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 64)(18 65)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 91)(89 92)(90 93)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 6A 6B 6C ··· 6N 6O 6P 6Q 6R 9A ··· 9F 9G ··· 9AD 18A ··· 18F 18G ··· 18AD 18AE ··· 18AP order 1 2 2 2 3 3 3 ··· 3 6 6 6 ··· 6 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 9 9 1 1 2 ··· 2 1 1 2 ··· 2 9 9 9 9 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 9 ··· 9

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 S3 D6 C3×S3 S3×C6 S3×C9 S3×C18 kernel C18×C3⋊S3 C9×C3⋊S3 C32×C18 C6×C3⋊S3 C3×C3⋊S3 C32×C6 C2×C3⋊S3 C3⋊S3 C3×C6 C3×C18 C3×C9 C3×C6 C32 C6 C3 # reps 1 2 1 2 4 2 6 12 6 4 4 8 8 24 24

Matrix representation of C18×C3⋊S3 in GL4(𝔽19) generated by

 8 0 0 0 0 8 0 0 0 0 13 0 0 0 0 13
,
 11 0 0 0 0 7 0 0 0 0 11 0 0 0 0 7
,
 11 0 0 0 0 7 0 0 0 0 7 0 0 0 0 11
,
 0 18 0 0 18 0 0 0 0 0 0 18 0 0 18 0
G:=sub<GL(4,GF(19))| [8,0,0,0,0,8,0,0,0,0,13,0,0,0,0,13],[11,0,0,0,0,7,0,0,0,0,11,0,0,0,0,7],[11,0,0,0,0,7,0,0,0,0,7,0,0,0,0,11],[0,18,0,0,18,0,0,0,0,0,0,18,0,0,18,0] >;

C18×C3⋊S3 in GAP, Magma, Sage, TeX

C_{18}\times C_3\rtimes S_3
% in TeX

G:=Group("C18xC3:S3");
// GroupNames label

G:=SmallGroup(324,143);
// by ID

G=gap.SmallGroup(324,143);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,68,2164,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^18=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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