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G = C6×D27order 324 = 22·34

Direct product of C6 and D27

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

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

C1C27 — C6×D27
C1C3C9C27C3×C27C3×D27 — C6×D27
C27 — C6×D27
C1C6

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

27C2
27C2
2C3
27C22
2C6
9S3
9S3
27C6
27C6
2C9
9D6
27C2×C6
2C18
3D9
3D9
9C3×S3
9C3×S3
2C27
3D18
9S3×C6
2C54
3C3×D9
3C3×D9
3C6×D9

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

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

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

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

90 conjugacy classes

class 1 2A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I9A···9I18A···18I27A···27AA54A···54AA
order1222333336666666669···918···1827···2754···54
size1127271122211222272727272···22···22···22···2

90 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6S3D6C3×S3D9S3×C6D18D27C3×D9D54C6×D9C3×D27C6×D27
kernelC6×D27C3×D27C3×C54D54D27C54C3×C18C3×C9C18C3×C6C9C32C6C6C3C3C2C1
# reps12124211232396961818

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

4600
0450
0045
,
100
0490
0089
,
10800
001
010
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

Export

Subgroup lattice of C6×D27 in TeX

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