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## G = C6×D27order 324 = 22·34

### Direct product of C6 and D27

Aliases: C6×D27, C543C6, C32.3D18, C273(C2×C6), (C3×C54)⋊2C2, C9.2(S3×C6), (C3×C9).7D6, (C3×C6).7D9, C6.5(C3×D9), C3.2(C6×D9), C18.5(C3×S3), (C3×C27)⋊3C22, (C3×C18).21S3, SmallGroup(324,65)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C6×D27
 Chief series C1 — C3 — C9 — C27 — C3×C27 — C3×D27 — C6×D27
 Lower central C27 — C6×D27
 Upper central C1 — C6

Generators and relations for C6×D27
G = < a,b,c | a6=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C6×D27
On 108 points
Generators in S108
(1 38 19 29 10 47)(2 39 20 30 11 48)(3 40 21 31 12 49)(4 41 22 32 13 50)(5 42 23 33 14 51)(6 43 24 34 15 52)(7 44 25 35 16 53)(8 45 26 36 17 54)(9 46 27 37 18 28)(55 89 64 98 73 107)(56 90 65 99 74 108)(57 91 66 100 75 82)(58 92 67 101 76 83)(59 93 68 102 77 84)(60 94 69 103 78 85)(61 95 70 104 79 86)(62 96 71 105 80 87)(63 97 72 106 81 88)
(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 97)(2 96)(3 95)(4 94)(5 93)(6 92)(7 91)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 108)(18 107)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 99)(27 98)(28 55)(29 81)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A ··· 9I 18A ··· 18I 27A ··· 27AA 54A ··· 54AA order 1 2 2 2 3 3 3 3 3 6 6 6 6 6 6 6 6 6 9 ··· 9 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 27 27 1 1 2 2 2 1 1 2 2 2 27 27 27 27 2 ··· 2 2 ··· 2 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 D6 C3×S3 D9 S3×C6 D18 D27 C3×D9 D54 C6×D9 C3×D27 C6×D27 kernel C6×D27 C3×D27 C3×C54 D54 D27 C54 C3×C18 C3×C9 C18 C3×C6 C9 C32 C6 C6 C3 C3 C2 C1 # reps 1 2 1 2 4 2 1 1 2 3 2 3 9 6 9 6 18 18

Matrix representation of C6×D27 in GL3(𝔽109) generated by

 46 0 0 0 45 0 0 0 45
,
 1 0 0 0 49 0 0 0 89
,
 108 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(109))| [46,0,0,0,45,0,0,0,45],[1,0,0,0,49,0,0,0,89],[108,0,0,0,0,1,0,1,0] >;

C6×D27 in GAP, Magma, Sage, TeX

C_6\times D_{27}
% in TeX

G:=Group("C6xD27");
// GroupNames label

G:=SmallGroup(324,65);
// by ID

G=gap.SmallGroup(324,65);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1443,381,5404,208,7781]);
// Polycyclic

G:=Group<a,b,c|a^6=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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