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G = Dic3×C27order 324 = 22·34

Direct product of C27 and Dic3

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: Dic3×C27, C3⋊C108, C6.C54, C54.4S3, C32.2C36, (C3×C27)⋊1C4, C2.(S3×C27), C6.8(S3×C9), (C3×C6).5C18, (C3×C54).1C2, (C3×C9).5C12, (C3×Dic3).C9, (C9×Dic3).C3, C18.10(C3×S3), (C3×C18).21C6, C3.4(C9×Dic3), C9.4(C3×Dic3), SmallGroup(324,11)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C27
C1C3C32C3×C9C3×C18C3×C54 — Dic3×C27
C3 — Dic3×C27
C1C54

Generators and relations for Dic3×C27
 G = < a,b,c | a27=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

2C3
3C4
2C6
2C9
3C12
2C18
2C27
3C36
2C54
3C108

Smallest permutation representation of Dic3×C27
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 57 10 66 19 75)(2 58 11 67 20 76)(3 59 12 68 21 77)(4 60 13 69 22 78)(5 61 14 70 23 79)(6 62 15 71 24 80)(7 63 16 72 25 81)(8 64 17 73 26 55)(9 65 18 74 27 56)(28 107 46 98 37 89)(29 108 47 99 38 90)(30 82 48 100 39 91)(31 83 49 101 40 92)(32 84 50 102 41 93)(33 85 51 103 42 94)(34 86 52 104 43 95)(35 87 53 105 44 96)(36 88 54 106 45 97)
(1 54 66 97)(2 28 67 98)(3 29 68 99)(4 30 69 100)(5 31 70 101)(6 32 71 102)(7 33 72 103)(8 34 73 104)(9 35 74 105)(10 36 75 106)(11 37 76 107)(12 38 77 108)(13 39 78 82)(14 40 79 83)(15 41 80 84)(16 42 81 85)(17 43 55 86)(18 44 56 87)(19 45 57 88)(20 46 58 89)(21 47 59 90)(22 48 60 91)(23 49 61 92)(24 50 62 93)(25 51 63 94)(26 52 64 95)(27 53 65 96)

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

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

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

162 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E9A···9F9G···9L12A12B12C12D18A···18F18G···18L27A···27R27S···27AJ36A···36L54A···54R54S···54AJ108A···108AJ
order123333344666669···99···91212121218···1818···1827···2727···2736···3654···5454···54108···108
size111122233112221···12···233331···12···21···12···23···31···12···23···3

162 irreducible representations

dim11111111111122222222
type+++-
imageC1C2C3C4C6C9C12C18C27C36C54C108S3Dic3C3×S3C3×Dic3S3×C9C9×Dic3S3×C27Dic3×C27
kernelDic3×C27C3×C54C9×Dic3C3×C27C3×C18C3×Dic3C3×C9C3×C6Dic3C32C6C3C54C27C18C9C6C3C2C1
# reps11222646181218361122661818

Matrix representation of Dic3×C27 in GL3(𝔽109) generated by

7800
070
007
,
10800
0450
010563
,
3300
07544
0134
G:=sub<GL(3,GF(109))| [78,0,0,0,7,0,0,0,7],[108,0,0,0,45,105,0,0,63],[33,0,0,0,75,1,0,44,34] >;

Dic3×C27 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{27}
% in TeX

G:=Group("Dic3xC27");
// GroupNames label

G:=SmallGroup(324,11);
// by ID

G=gap.SmallGroup(324,11);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,93,7781]);
// Polycyclic

G:=Group<a,b,c|a^27=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Dic3×C27 in TeX

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