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G = C3×Dic27order 324 = 22·34

Direct product of C3 and Dic27

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

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

C1C27 — C3×Dic27
C1C3C9C27C54C3×C54 — C3×Dic27
C27 — C3×Dic27
C1C6

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

2C3
27C4
2C6
2C9
9Dic3
27C12
2C18
2C27
3Dic9
9C3×Dic3
2C54
3C3×Dic9

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

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

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

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

90 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E9A···9I12A12B12C12D18A···18I27A···27AA54A···54AA
order123333344666669···91212121218···1827···2754···54
size11112222727112222···2272727272···22···22···2

90 irreducible representations

dim111111222222222222
type+++-+-+-
imageC1C2C3C4C6C12S3Dic3C3×S3D9C3×Dic3Dic9D27C3×D9Dic27C3×Dic9C3×D27C3×Dic27
kernelC3×Dic27C3×C54Dic27C3×C27C54C27C3×C18C3×C9C18C3×C6C9C32C6C6C3C3C2C1
# reps11222411232396961818

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

4500
0450
0045
,
10800
0260
010621
,
7600
05962
08150
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

Export

Subgroup lattice of C3×Dic27 in TeX

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