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## G = D9×C3×C6order 324 = 22·34

### Direct product of C3×C6 and D9

Aliases: D9×C3×C6, C93C62, C33.11D6, C183(C3×C6), (C3×C18)⋊14C6, (C32×C18)⋊3C2, C6.4(S3×C32), (C32×C9)⋊8C22, (C32×C6).18S3, C32.17(S3×C6), C3.1(S3×C3×C6), (C3×C9)⋊16(C2×C6), (C3×C6).37(C3×S3), SmallGroup(324,136)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — D9×C3×C6
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C32×D9 — D9×C3×C6
 Lower central C9 — D9×C3×C6
 Upper central C1 — C3×C6

Generators and relations for D9×C3×C6
G = < a,b,c,d | a3=b6=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 334 in 122 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C9, C9, C32, C32, C32, D6, C2×C6, D9, C18, C18, C3×S3, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C33, D18, S3×C6, C62, C3×D9, C3×C18, C3×C18, S3×C32, C32×C6, C32×C9, C6×D9, S3×C3×C6, C32×D9, C32×C18, D9×C3×C6
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2×C6, D9, C3×S3, C3×C6, D18, S3×C6, C62, C3×D9, S3×C32, C6×D9, S3×C3×C6, C32×D9, D9×C3×C6

Smallest permutation representation of D9×C3×C6
On 108 points
Generators in S108
(1 17 20)(2 18 21)(3 10 22)(4 11 23)(5 12 24)(6 13 25)(7 14 26)(8 15 27)(9 16 19)(28 40 52)(29 41 53)(30 42 54)(31 43 46)(32 44 47)(33 45 48)(34 37 49)(35 38 50)(36 39 51)(55 70 76)(56 71 77)(57 72 78)(58 64 79)(59 65 80)(60 66 81)(61 67 73)(62 68 74)(63 69 75)(82 97 103)(83 98 104)(84 99 105)(85 91 106)(86 92 107)(87 93 108)(88 94 100)(89 95 101)(90 96 102)
(1 41 23 32 14 50)(2 42 24 33 15 51)(3 43 25 34 16 52)(4 44 26 35 17 53)(5 45 27 36 18 54)(6 37 19 28 10 46)(7 38 20 29 11 47)(8 39 21 30 12 48)(9 40 22 31 13 49)(55 91 73 82 64 100)(56 92 74 83 65 101)(57 93 75 84 66 102)(58 94 76 85 67 103)(59 95 77 86 68 104)(60 96 78 87 69 105)(61 97 79 88 70 106)(62 98 80 89 71 107)(63 99 81 90 72 108)
(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 57)(2 56)(3 55)(4 63)(5 62)(6 61)(7 60)(8 59)(9 58)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 72)(18 71)(19 79)(20 78)(21 77)(22 76)(23 75)(24 74)(25 73)(26 81)(27 80)(28 88)(29 87)(30 86)(31 85)(32 84)(33 83)(34 82)(35 90)(36 89)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 99)(45 98)(46 106)(47 105)(48 104)(49 103)(50 102)(51 101)(52 100)(53 108)(54 107)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 6A ··· 6H 6I ··· 6Q 6R ··· 6AG 9A ··· 9AA 18A ··· 18AA order 1 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 9 9 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2 2 ··· 2

108 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 D9 C3×S3 D18 S3×C6 C3×D9 C6×D9 kernel D9×C3×C6 C32×D9 C32×C18 C6×D9 C3×D9 C3×C18 C32×C6 C33 C3×C6 C3×C6 C32 C32 C6 C3 # reps 1 2 1 8 16 8 1 1 3 8 3 8 24 24

Matrix representation of D9×C3×C6 in GL3(𝔽19) generated by

 11 0 0 0 1 0 0 0 1
,
 7 0 0 0 12 0 0 0 12
,
 1 0 0 0 17 16 0 0 9
,
 1 0 0 0 2 3 0 18 17
G:=sub<GL(3,GF(19))| [11,0,0,0,1,0,0,0,1],[7,0,0,0,12,0,0,0,12],[1,0,0,0,17,0,0,16,9],[1,0,0,0,2,18,0,3,17] >;

D9×C3×C6 in GAP, Magma, Sage, TeX

D_9\times C_3\times C_6
% in TeX

G:=Group("D9xC3xC6");
// GroupNames label

G:=SmallGroup(324,136);
// by ID

G=gap.SmallGroup(324,136);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,5404,208,7781]);
// Polycyclic

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

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