Copied to
clipboard

G = C6×C18order 108 = 22·33

Abelian group of type [6,18]

direct product, abelian, monomial

Aliases: C6×C18, SmallGroup(108,29)

Series: Derived Chief Lower central Upper central

C1 — C6×C18
C1C3C32C3×C9C3×C18 — C6×C18
C1 — C6×C18
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 2A2B2C3A···3H6A···6X9A···9R18A···18BB
order12223···36···69···918···18
size11111···11···11···11···1

108 irreducible representations

dim11111111
type++
imageC1C2C3C3C6C6C9C18
kernelC6×C18C3×C18C2×C18C62C18C3×C6C2×C6C6
# reps13621861854

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

1800
0110
001
,
1400
050
003
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

Export

Subgroup lattice of C6×C18 in TeX

׿
×
𝔽