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

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