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## G = C6×C18order 108 = 22·33

### Abelian group of type [6,18]

Aliases: C6×C18, SmallGroup(108,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6×C18
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C6×C18
 Lower central C1 — C6×C18
 Upper central C1 — C6×C18

Generators and relations for C6×C18
G = < a,b | a6=b18=1, ab=ba >

Smallest permutation representation of C6×C18
Regular action on 108 points
Generators in S108
(1 99 70 35 42 84)(2 100 71 36 43 85)(3 101 72 19 44 86)(4 102 55 20 45 87)(5 103 56 21 46 88)(6 104 57 22 47 89)(7 105 58 23 48 90)(8 106 59 24 49 73)(9 107 60 25 50 74)(10 108 61 26 51 75)(11 91 62 27 52 76)(12 92 63 28 53 77)(13 93 64 29 54 78)(14 94 65 30 37 79)(15 95 66 31 38 80)(16 96 67 32 39 81)(17 97 68 33 40 82)(18 98 69 34 41 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)

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

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

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

C6×C18 is a maximal subgroup of
C6.D18  C62.C9  C62.11C32  C62.12C32  C62.13C32  C62.14C32  C62.15C32  C62.16C32  C62.25C32

108 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 6A ··· 6X 9A ··· 9R 18A ··· 18BB order 1 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C3 C6 C6 C9 C18 kernel C6×C18 C3×C18 C2×C18 C62 C18 C3×C6 C2×C6 C6 # reps 1 3 6 2 18 6 18 54

Matrix representation of C6×C18 in GL3(𝔽19) generated by

 18 0 0 0 11 0 0 0 1
,
 14 0 0 0 5 0 0 0 3
G:=sub<GL(3,GF(19))| [18,0,0,0,11,0,0,0,1],[14,0,0,0,5,0,0,0,3] >;

C6×C18 in GAP, Magma, Sage, TeX

C_6\times C_{18}
% in TeX

G:=Group("C6xC18");
// GroupNames label

G:=SmallGroup(108,29);
// by ID

G=gap.SmallGroup(108,29);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,147]);
// Polycyclic

G:=Group<a,b|a^6=b^18=1,a*b=b*a>;
// generators/relations

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