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## G = C6×C9⋊S3order 324 = 22·34

### Direct product of C6 and C9⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C6×C9⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C3×C9⋊S3 — C6×C9⋊S3
 Lower central C3×C9 — C6×C9⋊S3
 Upper central C1 — C6

Generators and relations for C6×C9⋊S3
G = < a,b,c,d | a6=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 556 in 130 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2 [×2], C3 [×2], C3 [×3], C3 [×4], C22, S3 [×8], C6 [×2], C6 [×3], C6 [×6], C9 [×3], C9 [×3], C32 [×2], C32 [×3], C32 [×4], D6 [×4], C2×C6, D9 [×6], C18 [×3], C18 [×3], C3×S3 [×8], C3⋊S3 [×2], C3×C6 [×2], C3×C6 [×3], C3×C6 [×4], C3×C9, C3×C9 [×3], C3×C9 [×4], C33, D18 [×3], S3×C6 [×4], C2×C3⋊S3, C3×D9 [×6], C9⋊S3 [×2], C3×C18, C3×C18 [×3], C3×C18 [×4], C3×C3⋊S3 [×2], C32×C6, C32×C9, C6×D9 [×3], C2×C9⋊S3, C6×C3⋊S3, C3×C9⋊S3 [×2], C32×C18, C6×C9⋊S3
Quotients: C1, C2 [×3], C3, C22, S3 [×4], C6 [×3], D6 [×4], C2×C6, D9 [×3], C3×S3 [×4], C3⋊S3, D18 [×3], S3×C6 [×4], C2×C3⋊S3, C3×D9 [×3], C9⋊S3, C3×C3⋊S3, C6×D9 [×3], C2×C9⋊S3, C6×C3⋊S3, C3×C9⋊S3, C6×C9⋊S3

Smallest permutation representation of C6×C9⋊S3
On 108 points
Generators in S108
(1 62 33 67 43 48)(2 63 34 68 44 49)(3 55 35 69 45 50)(4 56 36 70 37 51)(5 57 28 71 38 52)(6 58 29 72 39 53)(7 59 30 64 40 54)(8 60 31 65 41 46)(9 61 32 66 42 47)(10 75 21 99 102 83)(11 76 22 91 103 84)(12 77 23 92 104 85)(13 78 24 93 105 86)(14 79 25 94 106 87)(15 80 26 95 107 88)(16 81 27 96 108 89)(17 73 19 97 100 90)(18 74 20 98 101 82)
(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 30 37)(2 31 38)(3 32 39)(4 33 40)(5 34 41)(6 35 42)(7 36 43)(8 28 44)(9 29 45)(10 108 24)(11 100 25)(12 101 26)(13 102 27)(14 103 19)(15 104 20)(16 105 21)(17 106 22)(18 107 23)(46 57 68)(47 58 69)(48 59 70)(49 60 71)(50 61 72)(51 62 64)(52 63 65)(53 55 66)(54 56 67)(73 87 91)(74 88 92)(75 89 93)(76 90 94)(77 82 95)(78 83 96)(79 84 97)(80 85 98)(81 86 99)
(1 74)(2 73)(3 81)(4 80)(5 79)(6 78)(7 77)(8 76)(9 75)(10 47)(11 46)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 94)(29 93)(30 92)(31 91)(32 99)(33 98)(34 97)(35 96)(36 95)(37 88)(38 87)(39 86)(40 85)(41 84)(42 83)(43 82)(44 90)(45 89)(64 104)(65 103)(66 102)(67 101)(68 100)(69 108)(70 107)(71 106)(72 105)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 6A 6B 6C ··· 6N 6O 6P 6Q 6R 9A ··· 9AA 18A ··· 18AA order 1 2 2 2 3 3 3 ··· 3 6 6 6 ··· 6 6 6 6 6 9 ··· 9 18 ··· 18 size 1 1 27 27 1 1 2 ··· 2 1 1 2 ··· 2 27 27 27 27 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 S3 D6 D6 C3×S3 D9 C3×S3 S3×C6 D18 S3×C6 C3×D9 C6×D9 kernel C6×C9⋊S3 C3×C9⋊S3 C32×C18 C2×C9⋊S3 C9⋊S3 C3×C18 C3×C18 C32×C6 C3×C9 C33 C18 C3×C6 C3×C6 C9 C32 C32 C6 C3 # reps 1 2 1 2 4 2 3 1 3 1 6 9 2 6 9 2 18 18

Matrix representation of C6×C9⋊S3 in GL5(𝔽19)

 8 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 11 0 0 0 0 0 11
,
 1 0 0 0 0 0 6 0 0 0 0 0 16 0 0 0 0 0 6 0 0 0 0 0 16
,
 1 0 0 0 0 0 7 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 1
,
 18 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(19))| [8,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,11],[1,0,0,0,0,0,6,0,0,0,0,0,16,0,0,0,0,0,6,0,0,0,0,0,16],[1,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1],[18,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C6×C9⋊S3 in GAP, Magma, Sage, TeX

C_6\times C_9\rtimes S_3
% in TeX

G:=Group("C6xC9:S3");
// GroupNames label

G:=SmallGroup(324,142);
// by ID

G=gap.SmallGroup(324,142);
# by ID

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

G:=Group<a,b,c,d|a^6=b^9=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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