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## G = C3×Dic27order 324 = 22·34

### Direct product of C3 and Dic27

Aliases: C3×Dic27, C273C12, C54.3C6, C6.4D27, C32.2Dic9, (C3×C27)⋊2C4, C2.(C3×D27), C6.2(C3×D9), (C3×C6).5D9, C18.3(C3×S3), (C3×C54).2C2, (C3×C18).18S3, C3.2(C3×Dic9), (C3×C9).5Dic3, C9.1(C3×Dic3), SmallGroup(324,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C3×Dic27
 Chief series C1 — C3 — C9 — C27 — C54 — C3×C54 — C3×Dic27
 Lower central C27 — C3×Dic27
 Upper central C1 — C6

Generators and relations for C3×Dic27
G = < a,b,c | a3=b54=1, c2=b27, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Dic27
On 108 points
Generators in S108
(1 19 37)(2 20 38)(3 21 39)(4 22 40)(5 23 41)(6 24 42)(7 25 43)(8 26 44)(9 27 45)(10 28 46)(11 29 47)(12 30 48)(13 31 49)(14 32 50)(15 33 51)(16 34 52)(17 35 53)(18 36 54)(55 91 73)(56 92 74)(57 93 75)(58 94 76)(59 95 77)(60 96 78)(61 97 79)(62 98 80)(63 99 81)(64 100 82)(65 101 83)(66 102 84)(67 103 85)(68 104 86)(69 105 87)(70 106 88)(71 107 89)(72 108 90)
(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 82 28 55)(2 81 29 108)(3 80 30 107)(4 79 31 106)(5 78 32 105)(6 77 33 104)(7 76 34 103)(8 75 35 102)(9 74 36 101)(10 73 37 100)(11 72 38 99)(12 71 39 98)(13 70 40 97)(14 69 41 96)(15 68 42 95)(16 67 43 94)(17 66 44 93)(18 65 45 92)(19 64 46 91)(20 63 47 90)(21 62 48 89)(22 61 49 88)(23 60 50 87)(24 59 51 86)(25 58 52 85)(26 57 53 84)(27 56 54 83)

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

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

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

90 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 9A ··· 9I 12A 12B 12C 12D 18A ··· 18I 27A ··· 27AA 54A ··· 54AA order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 9 ··· 9 12 12 12 12 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 1 1 2 2 2 27 27 1 1 2 2 2 2 ··· 2 27 27 27 27 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 C3 C4 C6 C12 S3 Dic3 C3×S3 D9 C3×Dic3 Dic9 D27 C3×D9 Dic27 C3×Dic9 C3×D27 C3×Dic27 kernel C3×Dic27 C3×C54 Dic27 C3×C27 C54 C27 C3×C18 C3×C9 C18 C3×C6 C9 C32 C6 C6 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 2 3 2 3 9 6 9 6 18 18

Matrix representation of C3×Dic27 in GL3(𝔽109) generated by

 45 0 0 0 45 0 0 0 45
,
 108 0 0 0 26 0 0 106 21
,
 76 0 0 0 59 62 0 81 50
G:=sub<GL(3,GF(109))| [45,0,0,0,45,0,0,0,45],[108,0,0,0,26,106,0,0,21],[76,0,0,0,59,81,0,62,50] >;

C3×Dic27 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{27}
% in TeX

G:=Group("C3xDic27");
// GroupNames label

G:=SmallGroup(324,10);
// by ID

G=gap.SmallGroup(324,10);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,1443,381,5404,208,7781]);
// Polycyclic

G:=Group<a,b,c|a^3=b^54=1,c^2=b^27,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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